Interferometer and method for measuring characteristics of optically unresolved surface features

ABSTRACT

Disclosed is an interferometry analysis method that includes comparing information derivable from multiple interferometry signals corresponding to different surface locations of a test object to information corresponding to multiple models of the test object, wherein the multiple models are parameterized by a series of characteristics that relate to one or more under-resolved lateral features of the test object; and outputting information about the under-resolved surface feature based on the comparison.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. 119(e) to U.S.Provisional Patent Application Ser. No. 60/737,016 filed Nov. 15, 2005and entitled “INTERFEROMETER AND METHOD FOR MEASURING CHARACTERISTICS OFOPTICALLY UNRESOLVED SURFACE FEATURES;” this application also claims thebenefit under 35 U.S.C. 120, and is a continuation-in-part, of U.S.patent application Ser. No. 10/795,579 filed Mar. 8, 2004 now U.S. Pat.No. 7,271,918 and entitled “PROFILING COMPLEX SURFACE STRUCTURES USINGSCANNING INTERFEROMETRY,” which in turn claims priority under 35 U.S.C.119(e) to the following U.S. Provisional Patent Applications: U.S.Patent Application Ser. No. 60/452,615 filed Mar. 6, 2003 and entitled“PROFILING COMPLEX SURFACE STRUCTURES USING HEIGHT SCANNINGINTERFEROMETRY,” U.S. Patent Application Ser. No. 60/452,465 filed Mar.6, 2003 and entitled “PROFILING COMPLEX SURFACE STRUCTURES USING SIGNALSFROM HEIGHT SCANNING INTERFEROMETRY,” and U.S. Patent Application Ser.No. 60/539,437 filed Jan. 26, 2004 and entitled “SURFACE PROFILING USINGAN INTERFERENCE PATTERN MATCHING TEMPLATE.” All of said relatedapplications are incorporated herein by reference.

BACKGROUND

The invention relates to using scanning interferometry to measuresurface topography and/or other characteristics of objects havingcomplex surface structures, such as thin film(s), discrete structures ofdissimilar materials, or discrete structures that are under-resolved bythe optical resolution of an interference microscope. Such measurementsare relevant to the characterization of flat panel display components,semiconductor wafer metrology, and in-situ thin film and dissimilarmaterials analysis.

Interferometric techniques are commonly used to measure the profile of asurface of an object. To do so, an interferometer combines a measurementwavefront reflected from the surface of interest with a referencewavefront reflected from a reference surface to produce aninterferogram. Fringes in the interferogram are indicative of spatialvariations between the surface of interest and the reference surface.

Typically, a scanning interferometer scans the optical path lengthdifference (OPD) between the reference and measurement legs of theinterferometer over a range comparable to, or larger than, the coherencelength of the interfering wavefronts, to produce a scanninginterferometry signal for each camera pixel used to measure theinterferogram. A limited coherence length can be produced, for example,by using a white-light source, which is referred to as scanning whitelight interferometry (SWLI). A typical scanning white lightinterferometry (SWLI) signal is a few fringes localized near the zerooptical path difference (OPD) position. The signal is typicallycharacterized by a sinusoidal carrier modulation (the “fringes”) withbell-shaped fringe-contrast envelope. The conventional idea underlyingSWLI metrology is to make use of the localization of the fringes tomeasure surface profiles.

SWLI processing techniques include two principle trends. The firstapproach is to locate the peak or center of the envelope, assuming thatthis position corresponds to the zero optical path difference (OPD) of atwo-beam interferometer for which one beam reflects from the objectsurface. The second approach is to transform the signal into thefrequency domain and calculate the rate of change of phase withwavelength, assuming that an essentially linear slope is directlyproportional to object position. See, for example, U.S. Pat. No.5,398,113 to Peter de Groot. This latter approach is referred to asFrequency Domain Analysis (FDA).

Unfortunately such assumptions may break down when applied to a testobject having a thin film because of reflections by the top surface andthe underlying film/substrate interface. Recently a method was disclosedin U.S. Pat. No. 6,545,763 to S. W. Kim and G. H. Kim to address suchstructures. The method fit the frequency domain phase profile of a SWLIsignal for the thin film structure to an estimated frequency domainphase profile for various film thicknesses and surface heights; Asimultaneous optimization determined the correct film thickness andsurface height.

Complex surface structures, e.g. patterned semiconductor wafers, may becomprised of features of dissimilar materials of various sizes from mmdown to a few tens of nm in size.

It is presently of considerable interest in the several industries,including in particular the semiconductor industry, to make quantitativemeasurements of surface topography. Due to the small size of typicalchip features, the instruments used to make these measurements typicallyshould have high spatial resolution both parallel and perpendicular tothe chip surface. Engineers and scientists use surface topographymeasuring systems for process control and to detect defects that occurin the course of manufacturing, especially as a result of processes suchas etching, polishing, cleaning and patterning.

Non-optical metrology tools such as top down critical dimension (CD)scanning electron microscopy (SEM) and atomic force microscopy (AFM) arein widespread use for obtaining pattern and topography information inthe semiconductor industry. While both of these techniques have therequired horizontal resolution they suffer from being extremely slow sothat it requires significant amounts of time to collect data over largeareas of the wafer. This is particularly true of the AFM. Top down CDSEM's are programmable and so can automatically collect data fromparticular regions of a set of wafers but even with this feature thetime required to collect full wafer data is prohibitive.

Conventional optical surface profilers such as confocal, interferometricor slope sensors overcome some of these difficulties; but generallybecome unusable when surface features are either too small, too closelyspaced or both, to be properly resolved and result in inaccurate surfaceheight variations.

Conventional interference microscopes measure surface profiles bydirectly associating interference phase with an optical path differencebetween a reference surface and a measurement surface. They have lateralresolution typically limited to approximately one wavelength of thesource illumination.

Scanning white light interference microscopes, also known as coherenceprobe microscopes, laser radar and vertical scanning interferometers,measure surface profiles take advantage of the limited coherence ofwhite light (or more generally, broadband) illumination to assist insurface profiling of discrete surface features, rough surface structuresand narrow lines. They have lateral resolution typically limited toapproximately one wavelength of the source illumination. Some of thesesystems may be arranged so as to measure the thickness of films.

Scatterometers determine a surface characteristic by matching thedistribution of scattered or diffracted light to a pre-computed libraryof the scattering and diffraction distributions from nominal structures.They do not directly measure surface profiles with respect to areference, as in an interferometer. Scatterometers also generally workonly with a limited set of 2D structures.

Confocal microscopes use a restricted depth of focus to section anobject vertically, so as to e.g. determine surface profile.

Nomarski microscopy and other differential techniques measuredifferences in surface height by comparing them to each other.

Ellipsometers measure the thin film and dissimilar material structure ofobjects using polarized light at high angles of incidence and theFresnel reflection coefficients. Generally the features of interest arelarge compared to the source wavelength and ellipsometers do not providesurface profile information.

SUMMARY

The inventors have realized that there is a wealth of information in ascanning interfometry signal, much of which is ignored in conventionalprocessing. While complex surface structures, such as thin films orunder-resolved surface features (i.e., lateral surface features smallerthan the spatial resolution of the interference microscope), may corruptconventional processing techniques based on identifying the location ofthe peak in the fringe contrast envelope or calculating a slope for thefrequency domain phase profile, new processing techniques disclosedherein can extract surface height information and/or information aboutthat the complex surface structure.

For example, while not assuming that the surface height information isdirectly related to the peak in the fringe contrast envelope, someembodiments of the invention assume that a change in surface heighttranslates the scanning interferometry signal with respect to areference scan position, but otherwise preserves the shape of thescanning interferometry signal. Thus, the shape of the scanninginterferometry signal is especially useful in characterizing complexsurface structure because it is independent of surface height.Similarly, in the frequency domain, some embodiments assume a change insurface height introduces a linear term in the frequency domain phaseprofile, even though the frequency domain profile itself may not belinear. However, the change in surface height leaves the frequencydomain amplitude profile unchanged. Therefore, the frequency domainamplitude profile is especially useful in characterizing complex surfacestructure.

After the complex surface structure is characterized, surface height canbe efficiently determined. For example, a cross-correlation between thescanning interferometry signal and a model signal having the shapecorresponding to the complex surface structure can produce a peak at ascan coordinate corresponding to the surface height. Similarly, in thefrequency domain, a phase contribution resulting from the complexsurface structure can be subtracted from the frequency domain phaseprofile and the surface height can be extracted using a conventional FDAanalysis.

Examples of complex surface structure include: simple thin films (inwhich case, for example, the variable parameter of interest may be thefilm thickness, the refractive index of the film, the refractive indexof the substrate, or some combination thereof); multilayer thin films;sharp edges and surface features that diffract or otherwise generatecomplex interference effects; unresolved surface roughness; unresolvedsurface features, for example, a sub-wavelength width groove on anotherwise smooth surface; dissimilar materials (for example, the surfacemay comprise a combination of thin film and a solid metal, in which casethe library may include both surface structure types and automaticallyidentify the film or the solid metal by a match to the correspondingfrequency-domain spectra); surface structure that give rise to opticalactivity such as fluorescence; spectroscopic properties of the surface,such as color and wavelength-dependent reflectivity;polarization-dependent properties of the surface; and deflections,vibrations or motions of the surface or deformable surface features thatresult in perturbations of the interference signal.

In some embodiments, the limited coherence length of the light used togenerate the scanning interferometry signal is based on a white lightsource, or more generally, a broadband light source. In otherembodiments, the light source may be monochromatic, and the limitedcoherence length can result from using a high numerical aperture (NA)for directing light to, and/or receiving light from, the test object.The high NA causes light rays to contact the test surface over a rangeof angles, and generates different spatial frequency components in therecorded signal as the OPD is scanned. In yet further embodiments, thelimited coherence can result from a combination of both effects.

The origin of the limited coherence length is also a physical basis forthere being information in the scanning interferometry signal.Specifically, the scanning interferometry signal contains informationabout complex surface structure because it is produced by light rayscontacting the test surface with many different wavelengths and/or atmany different angles.

In the processing techniques described herein, information derivablefrom a scanning interferometry signal for a first surface location of atest object (including the scanning interferometry signal itself) iscompared to information corresponding to multiple models of the testobject, where the multiple models are parameterized by a series ofcharacteristics for the test object. For example, the test object can bemodeled as a thin film and the series of characteristics can be a seriesof values for the thickness of the thin film. While the informationbeing compared might include, for example, information about thefrequency domain phase profile, it might also include information aboutthe shape of the scanning interferometry data and/or information aboutthe frequency domain amplitude profile. Furthermore, to focus thecomparison on the complex surface structure, and not the surface heightat the first surface location, the multiple models can all correspond toa fixed surface height for the test object at the first surfacelocation. The comparison itself can be based on calculating a meritfunction indicative of the similarity between the information from theactual scanning interferometry signal and the information from each ofthe models. For example, the merit function can be indicative of fitbetween the information derivable from the scanning interferometry dataand function parameterized by the series of characteristics.

Furthermore, in some embodiments, the series of characteristicscorresponds to a characteristic of the test object at second locationdifferent from the first location, including for example, diffractivesurface structures that contribute to the interface signal for the firstsurface locations. Thus, while we often refer to the complex surfacestructure as being something other than surface height at the firstsurface location corresponding to the scanning interferometry signal,the complex surface structure may correspond to surface height featuresspaced from the first surface location corresponding to the scanninginterferometry signal.

In yet further embodiments, the information derivable from the scanninginterferometry signal is an estimate for the relative surface height forthe first location. This information is compared to models whichcalculate what the apparent height would be for the first surfacelocation as measured by the interferometry system (taking into accountthe affect of under-resolved features at multiple surface locations) fordifferent values of the actual surface height profile. Based on thecomparison, one improves the accuracy of the surface height measurement,e.g., by selecting the actual surface heights used in the model thatproduced apparent heights most similar to those from the actualmeasurement.

More generally, information can be derived from the scanninginterferometry signals at multiple surface locations (even if at leastsome of the surface features present at those locations areunder-resolved), and this information can be compared to models whichcalculate what the information should look like as a function ofdifferent values parameterizing under-resolved features of interest. Forexample, the derivable information can correspond to an observed surfaceprofile determined using conventional scanning interferometryalgorithms. The under-resolved features of interest will be obscured inthis surface profile; however, they will still contribute to theobserved surface profile and information about the under-resolvedfeatures can be extracted from the observed surface profile by comparingit to different models.

For example, the individual lines of an under-resolved grating structurewill not be apparent in the surface profile observed from conventionalprocessing of the interferometry signals at multiple surface locations.Nonetheless, for example, the inventors have discovered that the actualmodulation depth of the grating structure can be correlated to thesurface height of the collective grating structure apparent in surfaceprofile obtained from conventionally processed scanning interferometrysignals.

The observed surface height of the collective grating structure is anexample of an “apparent” property of the test surface. In other words,it is example of a property related to how the test surface appears tothe measuring instrument because the test surface includesunder-resolved features. Such apparent properties can be compared todifferent models of the expected response of the instrument asparameterized by different values characterizing the under-resolvedfeatures of the test surface. The comparison can reveal which of thedifferent values yields the expected response most similar to theobserved response to thereby provide information about theunder-resolved feature of the test object based on the apparent propertyderived from the scanning interferometry signal(s).

The measurement technique and subsequent analysis approach describedherein is applicable to several semiconductor processing steps. With theuse of optical proximity corrections and/or phase shift masks,dimensions of patterned objects can be smaller than the wavelength usedby the optical lithography tool. For example, 193 nm lithography toolspattern 65 nm objects in today's high volume manufacturing facilitiesroutinely; the use of etch bias steps and hard mask structures canextend the lower limit to 45 nm and below. With the ability to print subwavelength features comes the need to monitor these features and theassociated etch and deposition steps. The embodiments of the currentdisclosure enable measurements of the nested patterned structures, wherenested is defined as a repeated surface structure of known shape whoseperiodicity is comparable to the lateral dimension of the structure. Inparticular, these nested structures could be used to monitor thefollowing process steps: isolation patterning and etch, poly-silicongate electrode patterning and etch, source/drain etch and deposition, aswell as many front end metallization patterning, etch, and polishprocesses. Additional applications include measurements of resist oversome film/substrate. Exposure and focus curves are characterized by linewidth and depth changes that are measurable with the inventive apparatusand method.

One example of such in-process metrology measurements of semiconductorchips include using scanning interferometry measurements for non-contactsurface topography measurements of semiconductor wafers during chemicalmechanical polishing (CMP) of a dielectric layer on the wafer. CMP isused to create a smooth surface for the dielectric layer, suitable forprecision optical lithography. Based on the results of theinterferometric topography methods, the process conditions for CMP(e.g., pad pressure, polishing slurry composition, etc.) can be adjustedto keep surface non-uniformities within acceptable limits.

We now summarize various aspects and features of the invention.

In general, in one aspect, a method is disclosed including: (i)comparing information derivable from multiple interferometry signalscorresponding to different surface locations of a test object toinformation corresponding to multiple models of the test object, whereinthe multiple models are parameterized by a series of characteristicsthat relate to one or more under-resolved lateral features of the testobject; and (ii) outputting information about the under-resolved surfacefeature based on the comparison.

Embodiments of the method may include any of the following features.

The one or more under-resolved lateral features of the test object maycorrespond to one or more of a pitch, a modulation depth, and an elementwidth for an under-resolved patterned lateral structure on the testobject. For example, the series of characteristics can include differentvalues for the modulation depth. Furthermore, the multiple models may berepresented by a correlation that maps possible outcomes for theinformation derivable from the multiple interferometry signals tocorresponding ones of the different values for the modulation depth, andthe comparing may be determining which of the different values for themodulation depth best corresponds to the information derivable from themultiple interferometry signals.

The modulation depth can be expressed relative to bias offset value.

At least some of the interferometry signals may be derived from anillumination of the test object whose polarization is oriented withrespect to the elements of the patterned lateral structure. For example,the polarization may be a linear polarization aligned orthogonal to thelength of the individual elements that define the patterned lateralstructure. (Referred to in the specification as “x-polarization.”)

The one or more under-resolved lateral features of the test object maycorrespond to one or more of a height and a position of a step on thetest object. For example, the series of characteristics may includedifferent values for the height or position of the step height.

The information derivable from the multiple interferometry signals mayinclude one or more values extracted from a height profile for the testobject derived from the multiple interferometry signals, wherein theunder-resolved surface feature is obscured or does not apparent in theextracted height profile. For example, the test object can include apatterned lateral structure whose individual element are obscured or donot apparent in the extracted surface height profile.

The information derivable from the multiple interferometry signals maybe a value for a height for a collection of under-resolved elements inthe patterned lateral structure extracted from the height profile. Theinformation about the under-resolved surface feature may correspond toone or more of a modulation depth and an element width for the patternedlateral structure.

The different surface locations for the interferometry signals mayinclude a reference portion of the test object providing a referenceheight value for the extracted height profile. For example, the testobject may be etched to produce the patterned structure, and thereference portion of the test object may be a portion of the test objectknown to not be etched.

At least some of the interferometry signals from which the heightprofile is determined may be derived from an illumination of the testobject whose polarization is oriented with respect to the elements ofthe patterned lateral structure. For example, the polarization may be alinear polarization aligned orthogonal to the length of the individualelements that define the patterned lateral structure (x-polarization).

The height profile may be obtained from a frequency domain analysis ofthe interferometry signals. Alternatively, the height profile may beobtained from a relative position of a coherence peak in eachinterferometry signal. The height profile may also be obtained usingother methods.

The under-resolved lateral features of the test object may have afeature size smaller than 400 nm, smaller than 200 nm, or even smallerthan 100 nm.

The models may be generated computationally using rigorous coupled waveanalysis (RCWA).

The models may be generated empirically from test objects having knownproperties.

The information about the under-resolved surface feature may beoutputted to a user.

The information about the under-resolved surface feature may beoutputted to an automated process control system for semiconductormanufacturing.

The interferometry signals may be scanning interferometry signals. Forexample, the scanning interferometry signal may be produced by imagingtest light emerging from the test object to interfere with referencelight on a detector, and varying an optical path length difference froma common source to the detector between interfering portions of the testand reference light, wherein the test and reference light are derivedfrom the common source, and wherein the scanning interferometry signalcorresponds to an interference intensity measured by the detector as theoptical path length difference is varied. The method may further includeproducing the scanning interferometry signal.

Such scanning interferometry signals may be low-coherence scanninginterferometry signals. For example, the test and reference light canhave a spectral bandwidth greater than 5% of a central frequency for thetest and reference light, and the optical path length difference isvaried over a range larger than the spectral coherence lengthcorresponding to that bandwidth to produce the scanning interferometrysignal. The low coherence may also result when the optics used to directtest light onto the test object and image it to the detector define anumerical aperture for the test light greater than 0.8. To reduce thecoherence length, the common source can be a spatially extended source.

In another related aspect, an apparatus is disclosed including: acomputer readable medium having a program that causes a processor in acomputer to compare information derivable from multiple interferometrysignals corresponding to different surface locations of a test object toinformation corresponding to multiple models of the test object, whereinthe multiple models are parameterized by a series of characteristicsthat relate to one or more under-resolved lateral features of the testobject, and output information about the under-resolved surface featurebased on the comparison.

In another related aspect, an apparatus is disclosed including: (i) aninterferometry system configured to produce multiple interferometrysignals corresponding to different surface locations of a test object;and (ii) an electronic processor coupled to the interferometry system toreceive the interferometry signals and programmed to compare informationderivable from the multiple interferometry signals to informationcorresponding to multiple models of the test object, wherein themultiple models are parameterized by a series of characteristics thatrelate to one or more under-resolved lateral features of the testobject, and output information about the under-resolved surface featurebased on the comparison.

Embodiments of the two apparatus may include any of the featuresdescribed above for the corresponding method.

In another aspect, a method is disclosed for determining one or morespatial properties of a grating structure on a test object, wherein thegrating structure includes line elements having widths smaller than 400nm so that the line elements are not fully resolved by an interferencemicroscope. The method including: (i) determining an apparent height fora collection of at least some of the grating lines from interferencesignals at different locations of the test object measured by theinterference microscope; (ii) providing an expected response for theinterference microscope for different possible values for the propertiesof the grating structure, wherein the expected response includescontributions from under-resolved line elements of the gratingstructure; (iii) comparing the apparent height to the expected responsefor the different possible values to determine information about thespatial properties of a grating structure; and (iv) outputting thedetermined information about the spatial properties of the gratingstructure.

In addition to features mentioned above, embodiments of this method mayinclude any of the following features.

The apparent height may be determined with reference to a referenceportion of the test object.

The interference microscope may illuminate the grating structure withlight polarized orthogonal to length of the individual grating lines(x-polarization) when determining the apparent height.

The determined information about the spatial properties of the gratingstructure may correspond to a modulation depth for the gratingstructure.

The grating structure may be a series of periodically spaced linesformed at least in part by etching portions of the test object betweenthe lines.

The interference signals may be scanning interference signals producedby imaging test light emerging from the test object to interfere withreference light on a detector, and varying an optical path lengthdifference from a common source to the detector between interferingportions of the test and reference light, wherein the test and referencelight are derived from the common source, and wherein each scanninginterferometry signal corresponds to an interference intensity measuredby the detector as the optical path length difference is varied. Forexample, the optical path length difference may be varied over a rangelarger than the coherence length of the interference microscope.

In a related aspect, an apparatus is disclosed for determining one ormore spatial properties of a grating structure on a test object, whereinthe grating structure includes line elements having widths smaller than400 nm so that the line elements are not fully resolved by aninterference microscope, the apparatus including: a computer readablemedium having a program that causes a processor in a computer to: 1)determine an apparent height for a collection of at least some of thegrating lines from interference signals at different locations of thetest object measured by the interference microscope; 2) provide anexpected response for the interference microscope for different possiblevalues for the properties of the grating structure, wherein the expectedresponse includes contributions from under-resolved line elements of thegrating structure; 3) compare the apparent height to the expectedresponse for the different possible values to determine informationabout the spatial properties of a grating structure; and 4) output thedetermined information about the spatial properties of the gratingstructure.

In another related aspect, an apparatus is disclosed for determining oneor more spatial properties of a grating structure on a test object,wherein the grating structure includes line elements having widthssmaller than 400 nm so that the line elements are not fully resolved byan interference microscope, the apparatus including: the interferencemicroscope; and an electronic processor coupled to the interferencemicroscope and programmed to: 1) determine an apparent height for acollection of at least some of the grating lines from interferencesignals at different locations of the test object measured by theinterference microscope; 2) provide an expected response for theinterference microscope for different possible values for the propertiesof the grating structure, wherein the expected response includescontributions from under-resolved line elements of the gratingstructure; 3) compare the apparent height to the expected response forthe different possible values to determine information about the spatialproperties of a grating structure; and 4) output the determinedinformation about the spatial properties of the grating structure.

Embodiments of the two apparatus may include any of the featuresdescribed above for the corresponding method.

In another aspect, a method is disclosed: (i) determining one or moreapparent properties of a test surface from interferometry signalsproduced by an interferometry system; (ii) comparing the apparentproperties determined from the interferometry signals to an expectedresponse for the interferometry system for different possible values forone or more under-resolved features of the test surface; and (iii)outputting information about the one or more under-resolved features ofthe test surface based on the comparison.

In addition to features mentioned above, embodiments of this method mayinclude any of the following features.

The interferometry system may be a scanning interferometry system.

The apparent properties of the test surface may be determined from theinterferometry signal based on variations in any of interference phase,interference contrast, and surface reflectivity.

The expected response may be calculated for variations in one or more ofsurface height and surface composition.

The test surface may include a patterned structure having a modulationdepth, a periodicity, and elements having a width, wherein the expectedresponse is calculated for variations in one or more of the modulationdepth, the periodicity, and element width. For example, the expectedresponse may be calculated for variations in the modulation depth.

The correspondence between the actual modulation depth and the apparentmodulation calculated for the expected response may include a positivecorrelation over a first range of actual modulation depths and anegative correlation over a second range of actual modulation depths.

The information about the one or more under-resolved features may beused to monitor one or more semiconductor processing steps, such as anyof isolation patterning and etch, poly-silicon gate electrode patterningand etch, source/drain etch and deposition, and metallizationpatterning, etch, and polish processing.

In another related aspect, an apparatus is disclosed including: acomputer readable medium having a program that causes a processor in acomputer to: 1) determine one or more apparent properties of a testsurface from interferometry signals produced by an interferometrysystem; 2) compare the apparent properties determined from theinterferometry signals to an expected response for the interferometrysystem for different possible values of one or more under-resolvedfeatures of the test surface; and 3) output information about the one ormore under-resolved features of the test surface based on thecomparison.

In another related aspect, an apparatus is disclosed including: aninterferometry system configured to produce multiple interferometrysignals corresponding to different surface locations of a test object;and an electronic processor coupled to the interferometry system toreceive the interferometry signals and programmed to: 1) determine oneor more apparent properties of a test surface from the interferometrysignals; 2) compare the apparent properties determined from theinterferometry signals to an expected response for the interferometrysystem for different possible values of one or more under-resolvedfeatures of the test surface; and 3) output information about the one ormore under-resolved features of the test surface based on thecomparison.

Embodiments of the two apparatus may include any of the featuresdescribed above for the corresponding method.

In another aspect, a method is disclosed including: (i) determining oneor more apparent properties of a test surface from an interferometrysignal (e.g., a scanning interferometry signal) produced by aninterferometry system; (ii) providing an expected response for theinterferometry system for different possible values of the properties ofthe test surface (for example, variations in one or more of surfaceheight and surface composition), wherein the expected response includescontributions from underresolved features of the test surface; and (iii)comparing the apparent properties determined from the interferometrysignal to the expected response for the different values of theproperties to improve the accuracy of the determined properties.

Embodiments of the method may include any of the following features.

The apparent properties of the test surface can be determining from theinterferometry signal based on variations in any of interference phase,interference contrast, and surface reflectivity.

The expected response can be calculated for variations in one or more ofsurface height and surface composition. For example, the test surfacecan include a patterned structure (e.g., a grating) having a modulationdepth, a periodicity, and elements having a width, and wherein theexpected response is calculated for variations in one or more of themodulation depth, the periodicity, and element width.

The correspondence between the apparent property associated with theinterferometry signal and the actual values of the property used togenerate models can include a positive correlation over a first range ofactual modulation depths and a negative correlation over a second rangeof actual modulation depths. For example, in a specific embodiment, theexpected response is calculated for variations in the modulation depth.In such cases, the correspondence between the actual modulation depthand the apparent modulation calculated for the expected response caninclude a positive correlation over a first range of actual modulationdepths and a negative correlation over a second range of actualmodulation depths.

In further embodiments, the underresolved feature can be single trench,step, or protrusion, rather than a series of such elements as in thepatterned structure. In such examples, the information from the actualsignal and for models can correspond to any of the depth (or height forthe case of the step or protrusion) and the location or width of thefeatures.

The method can further include using the improved accuracy of themeasured properties of the test surface to monitor semiconductorprocessing steps. For example, such processing steps can include any ofisolation patterning and etch, poly-silicon gate electrode patterningand etch, source/drain etch and deposition, and metallizationpatterning, etch, and polish processing.

In another aspect, an apparatus is disclosed including: a computerreadable medium having a program that causes a processor in a computerto compare apparent properties of a test surface determined from aninterferometry signal produced by an interferometry system to anexpected response for the interferometry system for different possiblevalues of the properties of the test surface, wherein the expectedresponse includes contributions from underresolved features of the testsurface, and improve the accuracy of the determined properties based onthe comparison.

In yet another aspect, an apparatus is disclosed including: aninterferometry system configured to produce an interferometry signal;and an electronic processor coupled to the interferometry system toreceive the interferometry signal and programmed to compare apparentproperties of a test surface determined from an interferometry signalproduced by an interferometry system to an expected response for theinterferometry system for different possible values of the properties ofthe test surface, wherein the expected response includes contributionsfrom underresolved features of the test surface, and improve theaccuracy of the determined properties based on the comparison.

Embodiments of such apparatus can further include features correspondingto any of those mentioned above in conjunction with the correspondingmethod. In general, in another aspect, the invention features a methodincluding: comparing information derivable from a scanninginterferometry signal for a first surface location of a test object toinformation corresponding to multiple models of the test object, whereinthe multiple models are parameterized by a series of characteristics forthe test object.

Embodiments of the invention may include any of the following features.

The method may further include determining an accurate characteristicfor the test object based on the comparison.

The method may further include determining a relative surface height forthe first surface location based on the comparison. Furthermore, thedetermining of the relative surface height may include determining whichmodel corresponds to an accurate one of the characteristic for the testobject based on the comparison, and using the model corresponding to theaccurate characteristic to calculate the relative surface height.

For example, the using of the model corresponding to the accuratecharacteristic may include compensating data from the scanninginterferometry signal to reduce contributions arising from the accuratecharacteristic. The compensating of the data may include removing aphase contribution arising from the accurate characteristic from a phasecomponent of a transform of the scanning interferometry signal for thetest object, and the using of the model corresponding to the accuratecharacteristic may further include calculating the relative surfaceheight from the phase component of the transform after the phasecontribution arising from the accurate characteristic has been removed.

In another example, using the model corresponding to the accuratecharacteristic to calculate the relative surface height may includedetermining a position of a peak in a correlation function used tocompare the information for the test object to the information for themodel corresponding to the accurate characteristic.

The method may further include comparing information derivable from thescanning interferometry signal for additional surface locations to theinformation corresponding to the multiple models. Also, the method mayfurther include determining a surface height profile for the test objectbased on the comparisons.

The comparing may include calculating one or more merit functionsindicative of a similarity between the information derivable from thescanning interferometry signal and the information corresponding to eachof the models.

The comparing may include fitting the information derivable from thescanning interferometry signal to an expression for the informationcorresponding to the models.

The information corresponding to the multiple models may includeinformation about at least one amplitude component of a transform (e.g.,a Fourier transform) of a scanning interferometry signal correspondingto each of the models of the test object. Likewise, the informationderivable from the scanning interferometry signal includes informationabout at least one amplitude component of a transform of the scanninginterferometry signal for the test object.

The comparing may include comparing a relative strength of the at leastone amplitude component for the test object to the relative strength ofthe at least one amplitude component for each of the models.

The information corresponding to the multiple models may be a functionof a coordinate for the transform. For example, the informationcorresponding to the multiple models may include an amplitude profile ofthe transform for each of the models. Furthermore, the comparing mayinclude comparing an amplitude profile of a transform of the scanninginterferometry signal for the test object to each of the amplitudeprofiles for the models.

The comparing may also include comparing information in a phase profileof the transform of the scanning interferometry signal for the testobject to information in a phase profile of the transform for each ofthe models. For example, the information in the phase profiles mayinclude information about nonlinearity of the phase profile with respectto the transform coordinate and/or information about a phase gap value.

The information derivable from the scanning interferometry signal andwhich is being compared may be a number. Alternatively, the informationderivable from the scanning interferometry signal and which is beingcompared may be a function. For example, it may be a function of scanposition or a function of spatial frequency.

The information for the test object may be derived from a transform(e.g., a Fourier transform) of the scanning interferometry signal forthe test object into a spatial frequency domain. The information for thetest object may include information about an amplitude profile of thetransform and/or a phase profile of the transform.

The information for the test object may relate to a shape of thescanning interferometry signal for the test object at the firstlocation. For example, the information for the test object may relate toa fringe contrast magnitude in the shape of the scanning interferometrysignal. It may also relate to a relative spacings between zero-crossingsin the shape of the scanning interferometry signal. It may also beexpressed as a function of scan position, wherein the function isderived from the shape of the scanning interferometry signal.

The comparing may include calculating a correlation function (e.g., acomplex correlation function) between the information for the testobject and the information for each of the models. The comparing mayfurther include determining one or more peak values in each of thecorrelation functions. The method may then further include determiningan accurate characteristic for the test object based on theparameterization of the model corresponding to the largest peak value.Alternately, or in addition, the method may further include determininga relative surface height for the test object at the first surfacelocation based on a coordinate for at least one of the peak values inthe correlation functions.

The multiple models may correspond to a fixed surface height for thetest object at the first location.

The series of characteristics may include a series of values for atleast one physical parameter of the test object. For example, the testobject may include a thin film layer having a thickness, and thephysical parameter may be the thickness of the thin film at the firstlocation.

The series of characteristics may include a series of characteristics ofthe test object at a second surface location different from the firstsurface location. For example, the test object may include structure atthe second surface location that diffracts light to contribute to thescanning interferometry signal for the first surface location. In oneexample, the series of characteristics at the second surface locationmay include permutations of a magnitude for a step height at the secondlocation and a position for the second location. In another example, theseries of characteristics at the second surface location may includepermutations of a modulation depth for a grating and an offset positionof the grating, wherein the grating extends over the second location.

Moreover, the information derivable from the interferometry signal cancorrespond to an estimate for the relative surface height of the firstsurface location. For example, the estimate of the relative of the firstsurface location can based on frequency domain analysis of theinterferometry signal, or the estimate of the relative surface height ofthe first surface location can be based on a relative position of acoherence peak in the interferometry signal.

The method can further include comparing information derivable from aninterferometry signal for one or more additional surface locations ofthe test object, including the second surface location, to theinformation corresponding to the multiple models of the test object.

For example, the information derivable from the interferometry signalfor the first and additional surface locations can correspond to asurface height profile for the test object for a range of surfacelocations including the first and additional surface locations. Also,the information corresponding to the multiple models can include asurface height profile expected to be produced by an interferometrysystem used to produce the interferometry signal when using conventionalprocessing of the interferometry signal for each of the multiple modelsof the test object, wherein the expected surface height profiles includecontributions from underresolved features that cause the interferencesignal for the first surface location to include contributions relatedto test object features at the second surface location. For example, thetest object can include structure at the second surface location thatdiffracts light to contribute to the interferometry signal for the firstsurface location.

In one example, the test object includes a patterned structure extendingover the first and additional surface locations, and the informationderivable from the scanning interferometry signal for the first andadditional surface locations includes an estimate for at least one of amodulation depth for the patterned structure, a periodicity for thepatterned structure, and a width for each element of the patternedstructure.

Likewise, for the example of the patterned structure, the series ofcharacteristics at the second surface location can include differentvalues for each of at least one of a modulation depth for the patternedstructure, a periodicity for the patterned structure, and a width foreach element of the patterned structure at the first surface location.

In further embodiments, the underresolved feature can be single trench,step, or protrusion, rather than a series of such elements as in thepatterned structure. In such examples, the information from the actualsignal and for models can correspond to any of the depth (or height forthe case of the step or protrusion) and the location or width of thefeatures.

The information derivable from the interferometry signal can correspondto an estimate for the relative surface height of the first surfacelocation, and the series of characteristics at the second surfacelocation can corresponds to series of values for a relative surfaceheight at the second surface location.

For example, the test object can include a patterned structure spanningthe first and second surface locations, and the estimate for therelative surface height of the first surface location corresponds to anestimate for a modulation depth of the patterned structure and theseries of values for the relative surface height at the second surfacelocation also correspond to the modulation depth for the patternedstructure. The multiple models can correlate different estimates for themodulation depth to corresponding ones of the values to improve theestimate for the modulation depth.

In some embodiments, the multiple models correlate the informationderivable from the interferometry signal to a corresponding value forthe information for the multiple models, wherein the correspondencevaries from a positive correlation to a negative correlation.

For example, for the case of the patterned structure, the correlationbetween the different estimates for the modulation depth and thecorresponding values for modulation depths from the models varies from apositive correlation to a negative correlation.

The method can further include determining an accurate characteristicfor the test object based on the comparison, such as a relative surfaceheight for the first surface location. For example, the determining ofthe relative surface height can include determining which modelcorresponds to an accurate one of the characteristic for the test objectbased on the comparison, and using the model corresponding to theaccurate characteristic to determine the relative surface height. Theseries of characteristics may be a series of surface materials for thetest object.

The series of characteristics may be a series of surface layerconfigurations for the test object.

The scanning interferometry signal may be produced by a scanninginterferometry system, and the comparing may include accounting forsystematic contributions to the scanning interferometry signal arisingfrom the scanning interferometry system. For example, the systematiccontributions may include information about a dispersion in a phasechange on reflection from components of the scanning interferometrysystem. Furthermore, the method may also include comparing informationderivable from the scanning interferometry signal for additional surfacelocations to the information corresponding to the multiple models, inwhich case, the systematic contributions may be resolved for multipleones of the surface locations. The method may further includecalibrating the systematic contributions of the scanning interferometrysystem using another test object having known properties.

The scanning interferometry signal may produced by imaging test lightemerging from the test object to interfere with reference light on adetector, and varying an optical path length difference from a commonsource to the detector between interfering portions of the test andreference light, wherein the test and reference light are derived fromthe common source (e.g., a spatially extended source), and wherein thescanning interferometry signal corresponds to an interference intensitymeasured by the detector as the optical path length difference isvaried.

The test and reference light may have a spectral bandwidth greater thanabout 5% of a central frequency for the test and reference light.

The common source may have a spectral coherence length, and the opticalpath length difference is varied over a range larger than the spectralcoherence length to produce the scanning interferometry signal.

Optics used to direct test light onto the test object and image it tothe detector may define a numerical aperture for the test light greaterthan about 0.8.

The method may further include producing the scanning interferometrysignal.

In another aspect, the invention features an apparatus including: acomputer readable medium having a program that causes a processor in acomputer to compare information derivable from a scanning interferometrysignal for a first surface location of a test object to informationcorresponding to multiple models for the test object, wherein themultiple models are parameterized by a series of characteristics for thetest object.

The apparatus may include any of the features described above inconnection with the method.

In another aspect, the invention features an apparatus including: ascanning interferometry system configured to produce a scanninginterferometry signal; and an electronic processor coupled to thescanning interferometry system to receive the scanning interferometrysignal and programmed to compare information derivable from a scanninginterferometry signal for a first surface location of a test object toinformation corresponding to multiple models of the test object, whereinthe multiple models are parameterized by a series of characteristics forthe test object.

The apparatus may include any of the features described above inconnection with the method.

In general, in another aspect, the invention features a methodincluding: chemically mechanically polishing a test object; collectingscanning interferometry data for a surface topography of the testobject; and adjusting process conditions for the chemically mechanicallypolishing of the test object based on information derived from thescanning interferometry data. For example, the process conditions may bepad pressure and/or polishing slurry composition. In preferredembodiments, adjusting the process conditions based on the informationderived from the scanning interferometry data may include comparinginformation derivable from the scanning interferometry signal for atleast a first surface location of a test object to informationcorresponding to multiple models of the test object, wherein themultiple models are parameterized by a series of characteristics for thetest object. Analysis of the scanning interferometry signal may furtherinclude any of the features described above with the first-mentionedmethod.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. In case of conflict withpublications, patent applications, patents, and other referencesmentioned incorporated herein by reference, the present specification,including definitions, will control.

For example, while scanning interferometry most frequently involvesmechanically scanning of the relative optical path lengths betweenreference and measurement legs, as used herein, scanning interferometryis intended to include additional non-mechanical means for varying therelative optical pathlength. For example, the interferometry signal canbe generating by varying the center wavelength of the light source overa range of wavelengths in an unequal path length interferometer (i.e.,where the nominal optical path length for the reference and measurementpaths differ.) The different wavelengths produce different phase shiftsfor the reference and measurement paths because they have differentoptical path lengths, thereby varying the phase shifts between thepaths.

In addition, while the term “light” can sometimes be understood to belimited to electromagnetic radiation in the visible spectrum, as usedherein, the term “light” is intended to include electromagneticradiation in any of ultraviolet, visible, near-infrared, and infraredspectral regions.

Other features, objects, and advantages of the invention will beapparent from the following detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of an interferometry method.

FIG. 2 is a flow chart showing a variation of the interferometry methodof FIG. 1.

FIG. 3 is a schematic drawing of a Linnik-type scanning interferometer.

FIG. 4 is a schematic drawing of a Mirau-type scanning interferometer.

FIG. 5 is a diagram showing illumination of the test sample through anobjective lens.

FIG. 6 shows theoretical Fourier amplitude spectra for scanninginterferometry data in two limits.

FIG. 7 shows two surface types, with and without a thin film

FIG. 8 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 0.

FIG. 9 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 50 nm.

FIG. 10 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 100 nm.

FIG. 11 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 300 nm.

FIG. 12 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 600 nm.

FIG. 13 shows the merit function search procedure for simulation of aSiO₂ film on a Si substrate with the thin film thickness being 1200 nm.

FIG. 14 shows the surface and substrate profiles determined for asimulation of a SiO₂ on Si thin film in which the film thickness variesuniformly from 0 to 1500 nm in 10-nm increments per pixel, with the topsurface always at zero.

FIG. 15 shows the surface and substrate profiles determined for asimulation identical to that in FIG. 14 except that random noise hasbeen added (2-bits rms out of an average 128 intensity bits).

FIG. 16 shows surface height profiles determined using conventional FDAanalysis (FIG. 16 a) and a library search method described herein (FIG.16 b) for a 2400 line per mm grating having an actual peak-to-valleymodulation depth of 120 nm.

FIG. 17 shows distortions caused by an under-resolved step height in ascanning interference signals for pixels corresponding to varioussurface locations near the step height.

FIG. 18 shows nonlinear distortions in the frequency domain phasespectra for pixels corresponding to surface locations to the left (FIG.18 a) and the right (FIG. 18 b) of the under-resolved step height inFIG. 17.

FIG. 19 shows surface height profiles determined using conventional FDAanalysis (FIG. 19 a) and a library search method described herein (FIG.1 b) for an under-resolved step height.

FIG. 20 shows an actual scanning interferometry signal of a base Sisubstrate without a thin film.

FIGS. 21 and 22 show interference template patterns for a bare Sisubstrate and a thin film structure with 1 micron of SiO2 on Si,respectively.

FIGS. 23 and 24 show the merit function as a function of scan positionsfor template functions in FIGS. 21 and 22, respectively.

FIG. 25 is a schematic diagram of an object with under-resolved surfacefeatures (a grating pattern).

FIG. 26 is a graph of the predicted surface profile of the gratingpattern in FIG. 25 based on a conventional interferometric analysis.

FIG. 27 is a graph of the apparent modulation depth of the grating inFIG. 25 for a conventional interferometric analysis as a function of itsactual modulation depth.

FIG. 28 is a graph showing theoretical and experimental instrumenttransfer functions of a typical white light scanning interferometer.

FIGS. 29 a-c show schematic diagrams of the a grating structure and oneembodiment of the under-resolved surface measurement technique disclosedherein.

FIG. 30 a shows a schematic of a model structure of 5 pure Si gratinglines (no top film layers), width W=120 nm and pitch L=320 nm. FIG. 30 bshows a rigorous coupled wave analysis (RCWA) showing the scanninginterferometry signal (in z-direction) for the model structure. This isfor a y polarization, parallel to the lines, which is more sensitive tothe tops of the lines than the areas.

FIGS. 31 a and 31 b show the scanning interferometry signal for thecenter pixel in FIG. 30 and for x- and y-polarization, respectively.

FIGS. 32 a and 32 b are surface height profiles derived from an FDAanalysis of the signals in FIG. 30 for the x-(orthogonal to gratinglines) and y-polarizations (parallel to grating lines), respectively.

FIGS. 33 a and 33 b are graphs showing the results of the RCWA analysisfor the apparent etch depth E′=H′−E as extracted from simulated scanninginterferometry data as a function of different actual etch depth E forthe y-polarization (parallel to grating lines) and x-(orthogonal tograting lines), respectively. The E′=E line is for reference.

FIGS. 34 a and 34 b are graphs showing the corresponding signal strengthfor the different actual etch depths E for the y-polarization (parallelto grating lines) and x-(orthogonal to grating lines), respectively.

FIG. 35 is an alternative graph of the same data as in FIG. 33 b thatshows the measurement bias or offset E−E′ as a function of the measuredstep height −E′ for the silicon grating.

FIG. 36 is a graph that shows the RCWA predictions for the measured etchdepth E′=H′−E as a function of the actual etch depth E for the silicongrating, with circularly polarized light, in comparison with aone-to-one correspondence line.

FIGS. 37 a and 37 b are graphs that show the RCWA predictions for themeasured (apparent) etch depth E′=H′−E as a function of the featurewidth W for an etch depth E=100 nm and pitch L=320 nm, for the five-linesilicon grating for y- and x-polarization, respectively. The solid linemarks the etch depth in both graphs, for reference.

FIG. 38 is a schematic diagram of an interferometry system showing howvarious components of the system can be under automated control.

Like reference numerals in different drawings refer to common elements.

DETAILED DESCRIPTION

FIG. 1 shows a flow chart that generally describes one embodiment of theinvention in which the analysis of the scanning interferometry data isperformed in the spatial frequency domain.

Referring to FIG. 1, to measure data from a test object surface aninterferometer is used to mechanically or electro-optically scan theoptical path difference (OPD) between a reference and measurement path,the measurement path being directed to an object surface. The OPD at thebeginning of the scan is a function of the local height of the objectsurface. A computer records an interference intensity signal during theOPD scan for each of multiple camera pixels corresponding to differentsurface locations of the object surface. Next, after storinginterference intensity signal as a function of OPD scan position foreach of the different surface locations, the computer performs atransform (e.g., a Fourier Transform) to generate a frequency-domainspectrum of the signal. The spectrum contains both magnitude and phaseinformation as a function of the spatial frequency of the signal in thescanning dimension. For example, a suitable frequency domain analysis(FDA) for generating such a spectrum is disclosed in commonly owned U.S.Pat. No. 5,398,113 by Peter de Groot and entitled “Method and Apparatusfor Surface Topography Measurements by Spatial-Frequency Analysis ofInterferograms,” the contents of which are incorporated herein byreference.

In a separate step, the computer generates a library of theoreticalpredictions for frequency-domain spectra for a variety of surfaceparameters and a model for the interferometer. These spectra may forexample cover a range of possible thin film thicknesses, surfacematerials, and surface textures. In preferred embodiments, the computergenerates library spectra for a constant surface height, e.g.height=zero. Thus, in such embodiments, the library contains noinformation regarding surface topography, only information relative tothe type of surface structure and the interaction of this surfacestructure, the optical system, the illumination, and detection systemwhen generating distinctive features of the frequency-domain spectra. Asan alternative, the prediction library may be generated empirically,using sample artifacts. As another alternative, the library may useinformation from prior supplemental measurements of the object surfaceprovided by other instruments, for example an ellipsometer, and anyother input from a user regarding known properties of the objectsurface, so as to reduce the number of unknown surface parameters. Anyof these techniques for library creation, theoretical modeling,empirical data, or theory augmented by supplemental measurements, may beexpanded by interpolation to generate intermediate values, either aspart of the library creation or in real time during a library search.

In a next step, the experimental data is compared to the predictionlibrary by means of a library search that provides surface structureparameters. In the example case of a film of unknown thickness, thelibrary for a single surface type, e.g. SiO₂ on Si, would range overmany possible film thicknesses with the top surface height always equalto zero. Another example case would be surface roughness, for which theadjustable parameter may be roughness depth and/or spatial frequency.The library search leads to a match to those characteristics of the FDAspectra that are independent of surface height, for example, the averagevalue of the magnitude spectrum, which is related to the overallreflectivity of the surface, or the variation in magnitude as a functionof spatial frequency, which in a monochromatic high-NA system relates tothe scattering angle of the reflected light.

The analysis may also include a system characterization, which includes,e.g. measuring one or more reference artifacts having a known surfacestructure and surface topography, so as to determine parameters such assystem wavefront error, dispersion, and efficiency that may not beincluded in the theoretical model.

Furthermore, the analysis may include an overall calibration, whichincludes e.g., measuring one or more reference artifacts to determinethe correlation between measured surface parameters, such as filmthickness as determined by the library search, and the values for theseparameters as determined independently, e.g. by ellipsometric analysis.

Based on the comparison of the experimental data to the predictionlibrary, the computer identifies the surface model corresponding to thebest match. It may then displays or transmits surface parameter resultsnumerically or graphically to the user or to a host system for furtheranalysis or for data storage. Using the surface parameter results, thecomputer may then determine surface height information in addition tocharacteristics identified by the library search. In some embodiments,the computer generates a compensated phase spectrum, for example bysubtracting the corresponding theoretical phase spectrum directly fromthe experimental phase spectrum. The computer then determines the localsurface height for one or more surface points by analysis of thecompensated phase as a function of spatial frequency, for example byanalysis of the coefficients generated by a linear fit. Thereafter, thecomputer generates a complete three-dimensional image constructed fromthe height data and corresponding image plane coordinates, together withgraphical or numerical display of the surface characteristics asdetermined by the library search.

In some cases, the library search and data collection can be performediteratively to further improve the results. Specifically, the librarysearch can be refined on a pixel-by-pixel or regional basis, by thecreation of refined libraries relevant to the local surface type. Forexample, if it is found that the surface has a thin film ofapproximately 1 micron during a preliminary library search, then thecomputer may generate a fine-grain library of example values close to 1micron to further refine the search.

In further embodiments, the user may only be interested in the surfacecharacteristics modeled by the prediction library, but not surfaceheight, in which case the steps for determining surface height are notperformed. Conversely, the user may only be interested in surfaceheight, but not the surface characteristics modeled in the predictionlibrary, in which case the computer uses the comparison between theexperimental data and the prediction library to compensate theexperimental data for the contributions of the surface characteristics,so that the surface height is more accurately determined, but need notexplicitly determine the surface characteristics or display them.

The analysis may be applied to a variety of surface analysis problems,including: simple thin films (in which case, for example, the variableparameter of interest may be the film thickness, the refractive index ofthe film, the refractive index of the substrate, or some combinationthereof); multilayer thin films; sharp edges and surface features thatdiffract or otherwise generate complex interference effects; unresolvedsurface roughness; unresolved surface features, for example, asub-wavelength width groove on an otherwise smooth surface; dissimilarmaterials (for example, the surface may comprise a combination of thinfilm and a solid metal, in which case the library may include bothsurface structure types and automatically identify the film or the solidmetal by a match to the corresponding frequency-domain spectra); opticalactivity such as fluorescence; spectroscopic properties of the surface,such as color and wavelength-dependent reflectivity;polarization-dependent properties of the surface; deflections,vibrations or motions of the surface or deformable surface features thatresult in perturbations of the interference signal; and data distortionsrelated to the data acquisition procedure, e.g. a data acquisitionwindow that does not fully encompass the interference intensity data.

The interferometer may include any of the following features: aspectrally narrow-band light source with a high numerical aperture (NA)objective; a spectrally broad band light source; a combination of a highNA objective and a spectrally broadband source; an interferometricmicroscope objectives, including oil/water immersion and solid immersiontypes, in e.g. Michelson, Mirau or Linnik geometries; a sequence ofmeasurements at multiple wavelengths; unpolarized light; and polarizedlight, including linear, circular, or structured. For example,structured polarized light may involve, for example, a polarizationmask, generating different polarizations for different segments of theillumination or imaging pupils, so as to reveal polarization-dependentoptical effects attributable to surface characteristics. Theinterferometer may also include the overall system calibration,described above.

In comparing the theoretical and experimental data, the library searchmay be based on any of the following: a product of, or a differencebetween, magnitude and/or phase data in the frequency spectrum,including, e.g., the product of, or difference between, the averagemagnitude and the average phase, the average magnitude itself, and theaverage phase itself; the slope, width and/or height of the magnitudespectrum; interference contrast; data in the frequency spectrum at DC orzero spatial frequency; nonlinearity or shape of the magnitude spectrum;the zero-frequency intercept of the phase; nonlinearity or shape of thephase spectrum; and any combination of these criteria. Note that as usedherein magnitude and amplitude are used interchangeably.

FIG. 2 shows a flow chart that generally describes another embodimentfor the analysis of scanning interferometry data. The analysis issimilar to that described for FIG. 1 except that comparison between theexperimental data and the prediction library is based on information inscan coordinate domain. The experimental signal may be characterized bya quasi-periodic carrier oscillation modulated in amplitude by anenvelope function with respect to the scan coordinate. In comparing thetheoretical and experimental data, the library search may be based onany of the following: average signal strength; the shape of the signalenvelope, including e.g. deviation from some ideal or reference shapesuch as a gaussian; the phase of the carrier signal with respect to theenvelope function; the relative spacing of zero crossings and/or signalmaxima and minima; values for maxima and minima and their ordering; peakvalue of the correlation between the library and measured signals, afteradjusting for optimal relative scan position; and any combination ofthese criteria.

Furthermore, in additional embodiments, the information compared to thelibrary models can be derived from experimental signals from multiplesurface locations. This can be especially useful when the test objectincludes under-resolved surface features because the experimentalinformation compared to the library models can correspond to acollective surface response of the test surface to interferometricmeasurement. For example, the information derived from the experimentalsignals from multiple surface locations can be a surface profile of thetest surface obtained from conventional processing of the interferometrysignals, or information derived from the surface profile. Suchprocessing yield only apparent properties of the test surface becausethe under-resolved features will be obscured. Nonetheless, theunder-resolved features can still leave signatures in the apparentsurface profile that can be correlated to more accurate informationabout the under-resolved features by comparing the apparent propertiesto corresponding models of the test object that are parameterized byvalues characterizing the under-resolved features.

Moreover, the test object can also include certain reference structuresthat can be involved in the comparison of the apparent features to thelibrary models. For example, when comparing the apparent surface heightprofile of a etched grating pattern whose individual lines areunder-resolved to various models, a portion of the test object that isknown to not be etched can provide a reference point for the apparentsurface height in the grating portion of the test object.

Thus, in certain embodiments, an interferometric profiler having both ameasurement and reference beam, for example a broadband or low coherenceinterferometer, is used to measure characteristics of a complex,unresolved surface structure such as may be found on a patternedsemiconductor wafer. The profiler interprets variations in interferencephase, contrast and/or surface reflectivity as variations in apparentmeasured surface height. In a separate step, data processing meanscalculate the expected response of the profile to possible variations inactual surface height and/or surface composition of the surface,including unresolved surface structures. Then, data processing meansdetermine a true surface characteristic by comparing the apparentmeasured surface height to the expected response of the profiler topossible variations in actual surface height, unresolved surfacestructure and/or surface composition.

Prior to, during or after the data acquisition above, the data processorcalculates the expected response of the instrument to possiblevariations in actual surface height and/or surface composition of thesurface, including unresolved surface structures. FIG. 25 is an exampleof an object having unresolved surface features for visible-wavelength(400-700 nm wavelength) interferometry. Specifically, FIG. 25 showsarray of elements (also referred to herein as a patterned structure or agrating structure). The unresolved surface features have a height Habove an adjacent surface S, a separation l and a width d. The height His also referred to herein as a modulation depth for the patternedstructure. By “unresolved” it is meant that the individual features arenot fully separated in the surface profile image, they have theincorrect profile and or the height H is incorrect, as a consequence ofthe limited lateral resolution of the instrument.

FIG. 26 illustrates a predicted response for the unresolved features onthe FIG. 25 object, for a visible-wavelength interferometer (560 nmcenter wavelength, 110 nm bandwidth full width half maximum) and anobjective NA of 0.8, using the Rayleigh Hypothesis technique (describedfurther below). The surface structures are unresolved, resulting in anapparent measured surface profile that does not resemble at all theactual surface structure. For this calculation, the height H=20 nm, theseparation I=200 nm and a width d=120 nm. FIG. 27 shows how theunresolved measured profile of FIG. 26 as a function of the actualheight H of the features. Noteworthy is that the relationship betweenthe measured and true height is complicated and even negativelycorrelated above 40 nm. This latter phenomenon can be explained as thedifficulty in coupling light into the narrow, sub-wavelength trenches

After data acquisition and after calculating the expected systemresponse, the data processor determines a true surface characteristic bycomparing the apparent measured surface height to the expected responseof the inventive apparatus to possible variations in actual surfaceheight, unresolved surface structure and/or surface composition.Following the example of FIG. 25 through FIG. 27, this involvesdetermining the relationship between the actual height H and theapparent measured height, as shown in FIG. 27, and determining theactual height from the measurement result using knowledge of thisrelationship. Alternative measurement modalities, such as determiningthe width d and separation l. As discussed further below, polarizing theobjective can also increase sensitivity to specific parameters such asetch depth, as can using an alternative light source to change theillumination wavelength and thereby adjust the sensitivity range.

Many processing techniques can be used to extract the apparent propertyor properties (e.g., apparent surface profile) from the scanninginterferometry data. For example, conventional techniques includeidentifying, for each pixel, the position corresponding to the peak orcenter of the fringe contrast envelope or to use, for each pixel,frequency domain analysis (FDA) and associate the rate of change ofphase with wavelength as directly proportional to surface height (see,for example, U.S. Pat. No. 5,398,133). In addition, it is also possibleto use more advanced processing techniques that try to remove thin filmeffects from the interferometry signals, such as disclosed in U.S.Patent Publication No. US-2005-0078318-A1 entitled “METHODS AND SYSTEMSFOR INTERFEROMETRIC ANALYSIS OF SURFACES AND RELATED APPLICATIONS,” byPeter de Groot or commonly owned U.S. patent application Ser. No.11/437,002 entitled “METHOD AND SYSTEM FOR ANALYZING LOW-COHERENCEINTERFEROMETRY SIGNALS FOR INFORMATION ABOUT THIN FILM STRUCTURES” byPeter de Groot and filed May 18, 2006, the contents of both applicationsbeing incorporated herein by reference. Of course, the model libraryused for comparison should take into account the processing techniqueused to extract the experimentally derived information. In what followswe provide a detailed mathematical description of the analyses andprovide examples. First, we describe exemplary scanning interferometers.Second, we determine a mathematical model for scanning interferometrydata. Third, we describe optical properties of surfaces and how to usesuch information to generate accurate models of scanning interferometrydata for different surface characteristics. Fourth, we describe howexperimental interferometry data can be compared to the predictionlibrary to provide information about the test object. Initially, we willdescribe thin film applications, and later we will describe applicationsto other complex surface structures, specifically, opticallyunder-resolved step heights and grating patterns. Also, we willinitially focus on analyses in the spatial frequency domain, and laterwe will describe analyses in the scan coordinate domain. Thereafter, wewill describe additional examples of techniques for extractinginformation about under-resolved surface features from theinterferometry signals such as in the example of FIGS. 25-27 above.

FIG. 3 shows a scanning interferometer of the Linnik type. Illuminationlight 102 from a source (not shown) is partially transmitted by a beamsplitter 104 to define reference light 106 and partially reflected bybeam splitter 104 to define measurement light 108. The measurement lightis focused by a measurement objective 110 onto a test sample 112 (e.g.,a sample comprising a thin single- or multi-layer film of one or moredissimilar materials). Similarly, the reference light is focused by areference objective 114 onto a reference mirror 116. Preferably, themeasurement and reference objectives have common optical properties(e.g., matched numerical apertures). Measurement light reflected (orscattered or diffracted) from the test sample 112 propagates backthrough measurement objective 110, is transmitted by beam splitter 104,and imaged by imaging lens 118 onto a detector 120. Similarly, referencelight reflected from reference mirror 116 propagates back throughreference objective 114, is reflected by beam splitter 104, and imagedby imaging lens 118 onto a detector 120, where it interferes with themeasurement light.

For simplicity, FIG. 3 shows the measurement and reference lightfocusing onto particular points on the test sample and reference mirror,respectively, and subsequently interfering on a corresponding point onthe detector. Such light corresponds to those portions of theillumination light that propagate perpendicular to the pupil planes forthe measurement and reference legs of the interferometer. Other portionsof the illumination light ultimately illuminate other points on the testsample and reference mirror, which are then imaged onto correspondingpoints on the detector. In FIG. 3, this is illustrated by the dashedlines 122, which correspond to the chief rays emerging from differentpoints on the test sample that are imaged to corresponding points on thedetector. The chief rays intersect in the center of the pupil plane 124of the measurement leg, which is the back focal plane of measurementobjective 110. Light emerging from the test sample at an angle differentfrom that of the chief rays intersect at a different location of pupilplane 124.

In preferred embodiments, detector 120 is a multiple element (i.e.,multi-pixel) camera to independently measure the interference betweenthe measurement and reference light corresponding to different points onthe test sample and reference mirror (i.e., to provide spatialresolution for the interference pattern).

A scanning stage 126 coupled to test sample 112 scans the position ofthe test sample relative to measurement objective 110, as denoted by thescan coordinate ζ in FIG. 3. For example, the scanning stage can bebased on a piezoelectric transducer (PZT). Detector 120 measures theintensity of the optical interference at one or more pixels of thedetector as the relative position of the test sample is being scannedand sends that information to a computer 128 for analysis.

Because the scanning occurs in a region where the measurement light isbeing focused onto the test sample, the scan varies the optical pathlength of the measurement light from the source to the detectordifferently depending on the angle of the measurement light incident on,and emerging from, the test sample. As a result, the optical pathdifference (OPD) from the source to the detector between interferingportions of the measurement and reference light scale differently withthe scan coordinate ζ depending on the angle of the measurement lightincident on, and emerging from, the test sample. In other embodiments ofthe invention, the same result can be achieved by scanning the positionof reference mirror 116 relative to reference objective 114 (instead ofscanning test sample 112 relative to measurement objective 110).

This difference in how OPD varies with the scan coordinate ζ introducesa limited coherence length in the interference signal measured at eachpixel of the detector. For example, the interference signal (as afunction of scan coordinate) is typically modulated by an envelopehaving a spatial coherence length on the order of λ/2(NA)², where λ isthe nominal wavelength of the illumination light and NA is the numericalaperture of the measurement and reference objectives. As describedfurther below, the modulation of the interference signal providesangle-dependent information about the reflectivity of the test sample.To increase the limited spatial coherence, the objectives in thescanning interferometer preferably define a large numerical aperture,e.g., greater than about 0.7 (or more preferably, greater than about0.8, or greater than about 0.9). The interference signal can also bemodulated by a limited temporal coherence length associated with thespectral bandwidth of the illumination source. Depending on theconfiguration of the interferometer, one or the other of these limitedcoherence length effects may dominate, or they may both contributesubstantially to the overall coherence length.

Another example of a scanning interferometer is the Mirau-typeinterferometer shown in FIG. 4.

Referring to FIG. 4, a source module 205 provides illumination light 206to a beam splitter 208, which directs it to a Mirau interferometricobjective assembly 210. Assembly 210 includes an objective lens 211, areference flat 212 having a reflective coating on a small centralportion thereof defining a reference mirror 215, and a beam splitter213. During operation, objective lens 211 focuses the illumination lighttowards a test sample 220 through reference flat 212. Beam splitter 213reflects a first portion of the focusing light to reference mirror 215to define reference light 222 and transmits a second portion of thefocusing light to test sample 220 to define measurement light 224. Then,beam splitter 213 recombines the measurement light reflected (orscattered) from test sample 220 with reference light reflected fromreference mirror 215, and objective 211 and imaging lens 230 image thecombined light to interfere on detector (e.g., a multi-pixel camera)240. As in the system of FIG. 3, the measurement signal(s) from thedetector is sent to a computer (not shown).

The scanning in the embodiment of FIG. 4 involves a piezoelectrictransducer (PZT) 260 coupled to Mirau interferometric objective assembly210, which is configured to scan assembly 210 as a whole relative totest sample 220 along the optical axis of objective 211 to provide thescanning interferometry data I(ζ,h) at each pixel of the camera.Alternatively, the PZT may be coupled to the test sample rather thanassembly 210 to provide the relative motion there between, as indicatedby PZT actuator 270. In yet further embodiments, the scanning may beprovided by moving one or both of reference mirror 215 and beam splitter213 relative to objective 211 along the optical axis of objective 211.

Source module 205 includes a spatially extended source 201, a telescopeformed by lenses 202 and 203, and a stop 204 positioned in the frontfocal plane of lens 202 (which coincides with the back focal plane oflens 203). This arrangement images the spatially extended to source ontothe pupil plane 245 of Mirau interferometric objective assembly 210,which is an example of Koehler imaging. The size of stop controls thesize of the illumination field on test sample 220. The system can alsoinclude an aperture stop (not shown) positioned between beam splitter208 and interference objective assembly 210. In other embodiments, thesource module may include an arrangement in which a spatially extendedsource is imaged directly onto the test sample which is known ascritical imaging. Either type of source module may be used with theLinnik-type scanning interferometry system of FIG. 3.

In further embodiments of the invention, the scanning interferometrysystem may used to determine angle-dependent scattering or diffractioninformation about a test sample, i.e., for scatterometry. For example,the scanning interferometry system may be used to illuminate a testsample with test incident over only a very narrow range of incidentangles (e.g., substantially normal incidence or otherwise collimated),which may then be scattered or diffracted by the test sample. The lightemerging from the sample is imaged to a camera to interfere withreference light as described above. The spatial frequency of eachcomponent in the scanning interferometry signal will depend vary withangle of the test light emerging from the test sample. Thus, a verticalscan (i.e., a scan along the optical axis of an objective) followed byFourier analysis allows for a measurement of diffracted and/or scatteredlight as a function of emerging angle, without directly accessing orimaging the back focal plane of the objective. To provide thesubstantially normal incidence illumination, for example, the sourcemodule can be configured to image a point source onto the pupil plane orto otherwise decrease the degree to which the illumination light fillsthe numerical aperature of the measurement objective. The scatterometrytechnique may be useful for resolving discrete structures in the samplesurface, such as grating lines, edges, or general surface roughness,which may diffract and/or scatter light to higher angles.

In much of the analysis herein, it is assumed that the polarizationstate of the light in the pupil plane is random, i.e., comprised ofapproximately equal amounts of both s polarizations (orthogonal to theplane of incidence) and p (orthogonal to the plane of incidence)polarizations. Alternative polarizations are possible, including pure spolarization, such as may be realized by means of a radial polarizerplaced in the pupil plane (e.g., in the back-focal plane of themeasurement object in the case of a Linnik interferometer and in theback focal plane of the common objective in the Mirau interferometer).Other possible polarizations include radial p polarization, circularpolarization, and modulated (e.g. two states, one following the other)polarization for ellipsometric measurements. In other words, opticalproperties of the test sample can be resolved not only with respect totheir angle- or wavelength-dependence, but also with respect to theirpolarization dependence or with respect to a selected polarization. Suchinformation may also be used to improve the accuracy of thin filmstructure characterization.

To provide such ellipsometry measurements, the scanning interferometrysystem may include a fixed or variable polarizer in the pupil plane.Referring again to FIG. 4, the Mirau-type interferometry system, forexample, includes polarization optics 280 in the pupil plane to select adesired polarization for the light incident on, and emerging from thetest sample. Furthermore, the polarization optics may be reconfigurableto vary the selected polarization. The polarization optics may includeone or more elements including polarizers, waveplates, apodizationapertures, and/or modulation elements for selecting a givenpolarization. Furthermore, the polarization optics may be fixed,structured or reconfigurable, for the purpose of generating data similarto that of an ellipsometer. For example, a first measurement with aradially-polarized pupil for s polarization, followed by aradially-polarized pupil for p polarization. In another example, one mayuse an apodized pupil plane with linearly polarized light, e.g., a slitor wedge, which can be rotated in the pupil plane so as to direct anydesired linear polarization state to the object, or a reconfigurablescreen such as a liquid crystal display.

Moreover, the polarization optics may provide a variable polarizationacross the pupil plane (e.g., by including multiple polarizers or aspatial modulator). Thus, one can “tag” the polarization state accordingto spatial frequency, for example, by providing a different polarizationfor high angles of incidence than shallow angles.

In yet further embodiments, the selectable polarization may be combinedwith a phase shift as a function of polarization. For example, thepolarization optics may include a linear polarizer is positioned in thepupil plane and followed by two waveplates (e.g., eighth-wave plates) inopposing quadrants of the pupil plane. The linear polarization resultsin a full range of polarization angles with respect to the incidentplanes of the objective. If the waveplates are aligned so that, forexample, the predominately s-polarized light has a fixed phase shift,then both radial s polarized and p polarized light are presentsimultaneously, but shifted in phase with respect to each other, e.g.,by pi, so that the interferometer is effectively detecting thedifference between these two polarization states as the fundamentalsignal.

In further embodiments, polarization optics may be positioned elsewherein the apparatus. For example, linear polarization can be achievedanywhere in the system.

We now describe a physical model for the scanning interferometry signal.

The object surface has height features h which we wish to profile overan area indexed by lateral coordinates x,y. The stage provides a smooth,continuous scan ζ either of the interference objective or, as shown, ofthe object itself. During the scan, a computer records intensity dataI_(ζ,h) for each image point or camera pixel in successive cameraframes. Note that the key dependencies of the intensity I_(ζ,h) on thescan position and surface height are indicated by subscripts—a notationthat we shall employ throughout.

A proper physical model of the optics can be very elaborate, taking intoaccount the partial coherence of the light source, polarization mixingin the interferometer, the imaging properties of high-NA objectives, andthe interaction of electric field vectors at high angles of incidenceand in the presence of discontinuous surface features.

For example, to rigorously predict the interferometer signal from agiven structure requires solving Maxwell's Equations for that structure.White light interferometry requires a sufficiently dense set ofwavelengths covering the bandwidth of the illumination. There are manyapproaches to solving Maxwell's Equations in 2D. In 2D one particularlysimple approach is based on the Rayleigh Hypothesis. In this approachthe structure is treated as a thin film stack but with the interfacesbetween each layer in the stack having a specific topography. TheRayleigh Hypothesis states that within each layer the electromagneticfield can be expanded in upward and downward propagating plane waves andthe solution can be generated by choosing the coefficients so that theelectromagnetic field satisfies the standard boundary conditions at theinterfaces. This approach is easy to implement, relatively fast and itgenerates the full optical transfer function or scattering matrix of thesurface at one wavelength and for one polarization in a singlecomputation. A limitation is that it generates valid solutions only incases where the topography of each interface covers less than roughlyhalf the wavelength in the layers bounding the interface. Hence, highindex materials such as silicon restrict the topography at visiblewavelengths to be much less than 100 nm.

3D modeling techniques include Finite Difference Time Domain (FDTD),Finite Element and rigorous coupled-wave analysis (RCWA). See, e.g., M.G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectricsurface-relief gratings.” J. Opt. Soc. Am., 72, 1385-1392, (1982), andM. Totzeck, “Numerical simulation of high-NA quantitative polarizationmicroscopy and corresponding near-fields”. Optik, 112 (2001) 381-390.Moreover, the Institute of Technical Optics (ITO) at the University ofStuttgart has developed software for performing RCWA called Microsim,based on the work of M. Totzek. These techniques are powerful althoughoften limited to small volumes, i.e., a few wavelengths on a side, tokeep both the memory requirements and the runtimes reasonable.Nonetheless, because such techniques can be used in advance to generatesuitable libraries, extensive computation time used to generate thelibraries does not hinder in-process application of the techniquesdisclosed herein.

For convenience, and to demonstrate certain aspects of the invention, wesimplify the model by assuming random polarization and diffuse,low-coherence extended sources. Modeling the interference signalsimplifies to adding up the contributions of all of the ray bundlespassing through the pupil plane of the objective and reflecting from theobject surface at an incident angle ψ, as shown in FIG. 5.

The interference contribution for a single ray bundle through theoptical system is proportional to

$\begin{matrix}{g_{\beta,k,\zeta,h} = {R_{\beta,k} + Z_{\beta,k} + {2\sqrt{R_{\beta,k}Z_{\beta,k}}{{\cos\left\lbrack {{2\beta\; k\;{n_{0}\left( {h - \zeta} \right)}} + \left( {\upsilon_{\beta,k} - \omega_{\beta,k}} \right)} \right\rbrack}.}}}} & (1)\end{matrix}$Where Z_(β,k) is the effective object intensity reflectivity, includinge.g. the effects of the beamsplitter, and R_(β,k) is the effectivereference reflectivity, including both the beamsplitter and thereference mirror. The index of the ambient medium is n₀, the directionalcosine for an incident angle ψ isβ=cos(ψ)  (2)and the wavenumber for the source illumination isk=(2π/λ)  (3)The sign convention for the phase causes an increase in surface heightto correspond to a positive change in phase. The phase term has acontribution ω_(β,k) for the object path in the interferometer,including thin film effects from the object surface, and a contributionυ_(β,k) for the reference path, including the reference mirror and otheroptics in the objective.

The total interference signal integrated over the pupil plane isproportional to

$\begin{matrix}{I_{\zeta,h} = {\int_{0}^{\infty}{\int_{0}^{1}{g_{\beta,k,\zeta,h}U_{\beta}V_{k}\beta\ {\mathbb{d}\beta}\ {\mathbb{d}k}}}}} & (4)\end{matrix}$where U_(β) is the pupil plane light distribution and V_(k) the opticalspectrum distribution. The weighting factor β in Eq. (4) follows from acos (ψ) term attributable to the projection angle and a sin (ψ) term forthe diameter of the annulus of width dψ in the pupil plane:cos(ψ)sin(ψ)dψ=−βdβ  (5)Here we assume that the objective obeys the Abbé sine condition as shownin FIG. 5. This relatively simple weighting is possible for randomlypolarized, spatially incoherent illumination, for which all ray bundlesare independent from each other. Finally, the integration limits overall incident angles implies 0≦β≦1 and the spectrum integration over allwavenumbers 0≦k≦∞.

In a frequency domain analysis (FDA), we first calculate the FourierTransform of the interference intensity signal I_(ζ,h). For the literal(non-numerical) analysis we shall use the un-normalized Fourier integral

$\begin{matrix}{q_{K,h} = {\int_{- \infty}^{\infty}{I_{\zeta,h}{\exp\left( {{\mathbb{i}}\; K\;\zeta} \right)}\ {\mathbb{d}\zeta}}}} & (6)\end{matrix}$where K is the spatial frequency, e.g. in cycles per μm. Thefrequency-domain value q_(K,h) carries units of inverse wavenumber, e.g.μm. From there follows a power spectrumQ _(K,h) =|q _(K,h)|²  (7)and a phase spectrumφ″_(K,h)=arg(q _(K,h)).  (8)The double prime for φ″_(K,h) means that there is a two-fold uncertaintyin the fringe order, both from pixel to pixel and overall with respectto the starting point in the scan. Conventional FDA then proceedsdirectly to a determination of surface topography by a linear fit to thephase spectrum φ″_(K,h) weighted by the power spectrum Q_(K,h). The fitprovides for each pixel a slopeσ_(h) ≈dφ″/dK  (9)and an interceptA″≈φ″ _(K=0,h).  (10)Note that the intercept or “phase gap” A″ is independent of height h,but carries the double prime inherited from the fringe order uncertaintyin the phase data. The slope σ is free of this uncertainty. From theintercept A″ and the slope σ_(h), we can define for a specific mean ornominal spatial frequency K0 a “coherence profile”Θ_(h)=σ_(h) K0  (11)and a “phase profile”θ″_(h)=Θ_(h) +A″.  (12)For the simple, idealized case of a perfectly uniform, homogeneousdielectric surface free of thin films and dissimilar material effects,and an optical system perfectly balanced for dispersion, the phase andcoherence profiles are linearly proportional to surface height:h _(Θ)=Θ_(h) /K0  (13)h″ _(θ)=θ″_(h) /K0  (14)Of the two height calculations, the height value h″_(θ) based on phaseis the more accurate, but it has the uncertainty in the fringe ordercharacteristic of monochromatic interferometry. For high resolution, wecan use the unambiguous but less precise height value h_(Θ) based oncoherence to remove this uncertainty and yield a final value h_(θ).

Conventional FDA assumes that even for less idealized situations, theinterference phase φ″_(K,h) is still nearly a linear function of spatialfrequency. This corresponds to conventional FDA processing of thescanning interferometry signal to get surface height. For the presentembodiment, however, we determine key parameters of the surfacestructure such as film thickness by comparing experimental data to atheoretical prediction that may include highly nonlinear phase spectraand associated modulations of the power spectrum.

To this end, we combine the definition of the Fourier Transform Eq. (6)with the interference signal Eq. (4) into the following formula for thepredicted FDA spectrum:

$\begin{matrix}{q_{K,h} = {\int_{- \infty}^{\infty}{\int_{0}^{\infty}{\int_{0}^{1}{g_{\beta,k,\zeta,h}{\exp\left( \ {{\mathbb{i}}\; K\;\zeta} \right)}U_{\beta}V_{k}\beta\ {\mathbb{d}\beta}\ {\mathbb{d}k}\ {\mathbb{d}\zeta}}}}}} & (15)\end{matrix}$To improve computational efficiency, a partial literal evaluation of thetriple integration in Eq. (15) can be performed.

The literal analysis of Eq. (15) begins with a change of the order ofintegration to first evaluate the individual interference signalsg_(β,k,ζ,h) over all scan positions ζ at fixed β and k:

$\begin{matrix}{q_{K,h} = {\int_{0}^{\infty}{\int_{0}^{1}{U_{\beta}V_{k}\beta\left\{ {\int_{- \infty}^{\infty}{g_{\beta,k,\zeta,h}{\exp\left( \ {{\mathbb{i}}\; K\;\zeta} \right)}\ {\mathbb{d}\zeta}}} \right\}\ {\mathbb{d}\beta}\ {{\mathbb{d}k}.}}}}} & (16)\end{matrix}$After expansion of the cosine term in g_(β,k,ζ,h) in the usual way using2 cos(u)=exp(iu)+exp(−iu),  (17)the inner integral over ζ evaluates to

$\begin{matrix}{{\int_{- \infty}^{\infty}{g_{\beta,k,\zeta,h}{\exp\left( \ {{\mathbb{i}}\; K\;\zeta} \right)}\ {\mathbb{d}\zeta}}} =_{K}{{\left( {R_{\beta,k} + Z_{\beta,k}} \right)\mspace{14mu}\ldots} + {\delta_{({K - {2\;\beta\; k\; n_{0}}})}\sqrt{R_{\beta,k}Z_{\beta,k}}{\exp\left\lbrack {{{\mathbb{i}2}\;\beta\; k\; n_{0}h} + {{\mathbb{i}}\left( {\upsilon_{\beta,k} - \omega_{\beta,k}} \right)}} \right\rbrack}\mspace{14mu}\ldots} + {\delta_{({K + {2\beta\; k\; n_{0}}})}\sqrt{R_{\beta,k}Z_{\beta,k}}{\exp\left\lbrack {{{- {\mathbb{i}2}}\;\beta\; k\; n_{0}h} - {{\mathbb{i}}\left( {\upsilon_{\beta,k} - \omega_{\beta,k}} \right)}} \right\rbrack}}}} & (18)\end{matrix}$where we have used

$\begin{matrix}{\delta_{K} = {\int_{- \infty}^{\infty}{{\exp\left( {{\mathbb{i}}\; K\;\zeta} \right)}\ {\mathbb{d}\zeta}}}} & (19) \\{\delta_{({K \pm {2\;\beta\; k\; n_{0}}})} = {\int_{- \infty}^{\infty}{{\exp\left( {{\mathbb{i}}\; K\;\zeta} \right)}\ {\exp\left( {{\pm {\mathbb{i}}}\; 2\;\beta\; k\; n_{0}\zeta} \right)}{{\mathbb{d}\zeta}.}}}} & (20)\end{matrix}$The δ function carries with it the inverse physical units of theargument, in this case, an inverse wavenumber.

These delta functions validate an equivalency between the spatialfrequency K and the product 2βkn₀. A logical change of variables for thenext integration is thereforeβ={circumflex over (κ)}/2kn ₀  (21)dβ=d{circumflex over (κ)}/2kn ₀  (22)where {circumflex over (κ)} has the same meaning as the spatialfrequency K, but will be used as a free integration variable. Eq. (18)can be written

$\begin{matrix}{{q_{K,h} = {{\int_{0}^{\infty}{\int_{0}^{2\; k\; n_{0}}{{\delta_{K}\left( {R_{\hat{\kappa},k} + Z_{\hat{\kappa},k}} \right)}\ \Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}\ {\mathbb{d}k}\mspace{20mu}\ldots}}} + {\int_{0}^{\infty}{\int_{0}^{2\; k\; n_{0}}{\delta_{({K - \hat{\kappa}})}\sqrt{R_{\hat{\kappa},k} + Z_{\hat{\kappa},k}}{\exp\left\lbrack {{{\mathbb{i}}\hat{\kappa}h} + {{\mathbb{i}}\left( {\upsilon_{\hat{\kappa},k} - \omega_{\hat{\kappa},k}} \right)}} \right\rbrack}\Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}{\mathbb{d}k}\mspace{20mu}\ldots}}} + {\int_{0}^{\infty}{\int_{0}^{2\; k\; n_{0}}{\delta_{({K - \hat{\kappa}})}\sqrt{R_{\hat{\kappa},k} + Z_{\hat{\kappa},k}}{\exp\left\lbrack {{{- {\mathbb{i}}}\hat{\kappa}\; h} - {{\mathbb{i}}\left( {\upsilon_{\hat{\kappa},k} - \omega_{\hat{\kappa},k}} \right)}} \right\rbrack}\Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}{\mathbb{d}k}}}}}}\mspace{20mu}{where}} & (23) \\{\mspace{20mu}{\Gamma_{\hat{\kappa},k} = {\frac{U_{\hat{\kappa},k}V_{k}\hat{\kappa}}{4\; k^{2}n_{0}^{2}}.}}} & (24)\end{matrix}$Note that by virtue of the change in variables, the β-dependence for theR, Z, υ, ω terms in Eq. (23) becomes a dependence upon {circumflex over(κ)} and k.

For the next step, we first note that

$\begin{matrix}{{\int_{0}^{2\; k\; n_{0}}{\delta_{K}f_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}}} = {\delta_{K}{\int_{0}^{\infty}{H_{({{2\; k\; n_{0}} - \hat{\kappa}})}f_{\hat{\kappa},k}\ {\mathbb{d}\hat{\kappa}}}}}} & (25) \\{{\int_{0}^{2\; k\; n_{0}}{\delta_{({K - \hat{\kappa}})}f_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}}} = {f_{K,k}H_{K}H_{({{2\; k\; n_{0}} - K})}}} & (26) \\{{\int_{0}^{2\; k\; n_{0}}{\delta_{({K - \hat{\kappa}})}f_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}}} = {f_{{- K},k}H_{- K}H_{({{2\; k\; n_{0}} + K})}}} & (27)\end{matrix}$where H is the unitless Heaviside step function defined by

$\begin{matrix}{H_{u} = \left\{ \begin{matrix}{{0\mspace{14mu}{for}\mspace{14mu} u} < 0} \\{1\mspace{14mu}{otherwise}}\end{matrix} \right.} & (28)\end{matrix}$and ƒ is an arbitrary function of K and k. Using Eqs. (25) through (27),Eq. (23) becomes

$\begin{matrix}{q_{K,h} = {{\delta_{K}{\int_{0}^{\infty}{\int_{0}^{\infty}{{H_{({{2\; k\; n_{0}} - \hat{\kappa}})}\ \left( {R_{\hat{\kappa},k} + Z_{\hat{\kappa},k}} \right)}\Gamma_{\hat{\kappa},k}{\mathbb{d}\hat{\kappa}}\ {\mathbb{d}k}\;\ldots}}}} + {\int_{0}^{\infty}{H_{K}H_{({{2\; k\; n_{0}} - K})}\sqrt{R_{K,k} + Z_{K,k}}\ {\exp\left\lbrack {{{\mathbb{i}}\; K\; h} + {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{K,k}} \right)}} \right\rbrack}\Gamma_{K,k}{\mathbb{d}\; k}\;\ldots}} + {\int_{0}^{\infty}{H_{- K}H_{({{2k\; n_{0}} + K})}\sqrt{R_{{- K},k} + Z_{{- K},k}}\ {\exp\left\lbrack {{{\mathbb{i}K}\; h} - {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{{- K},k}} \right)}} \right\rbrack}\Gamma_{{- K},k}{{\mathbb{d}k}.}}}}} & (29)\end{matrix}$Now using

$\begin{matrix}{{\int_{0}^{\infty}{\int_{0}^{\infty}{H_{({{2k\; n_{0}} - \hat{\kappa}})}\ {\mathbb{d}\hat{\kappa}}\ {\mathbb{d}k}}}} = {\int_{0}^{\infty}{\int_{0}^{\infty}{H_{({{2k\; n_{0}} - \hat{\kappa}})}\ f_{\hat{\kappa},k}{\mathbb{d}k}\ {\mathbb{d}\hat{\kappa}}}}}} & (30) \\{{\int_{0}^{\infty}{H_{K}H_{({{2k\; n_{0}} - K})}\ f_{K,k}{\mathbb{d}k}}}\  = {H_{K}{\int_{{K/2}\; n_{0}}^{\infty}{f_{K,k}\ {\mathbb{d}k}}}}} & (31) \\{{\int_{0}^{\infty}{H_{- K}H_{({{2k\; n_{0}} + K})}\ f_{{- K},k}{\mathbb{d}k}}}\  = {H_{- K}{\int_{{{- K}/2}\; n_{0}}^{\infty}{f_{{- K},k}\ {\mathbb{d}k}}}}} & (32)\end{matrix}$we have the final result

$\begin{matrix}{q_{K,k} = {{\delta_{K}{\int_{\kappa = 0}^{\infty}{\int_{{\hat{\kappa}/2}n_{0}}^{\infty}{\left( {R_{\kappa,k} + Z_{\kappa,k}} \right)\Gamma_{\hat{\kappa},k}\ {\mathbb{d}k}\ {\mathbb{d}\hat{\kappa}}}}}} + {H_{K}{\exp\left( {{\mathbb{i}K}\; h} \right)}{\int_{{K/2}n_{0}}^{\infty}{\sqrt{R_{K,k}Z_{K,k}}{\exp\left\lbrack {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{K,k}} \right)} \right\rbrack}\Gamma_{K,k}{\mathbb{d}\; k}\;\ldots}}} + {H_{{- K}\;}{\exp\left( {{\mathbb{i}}\; K\; h} \right)}{\int_{{{- K}/2}n_{0}}^{\infty}{\sqrt{R_{{- K},k}Z_{{- K},k}}{\exp\left\lbrack {- {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{{- K},k}} \right)}} \right\rbrack}\Gamma_{{- K},k}{\mathbb{d}k}}}}}} & (33)\end{matrix}$Because there are fewer integrations, Eq. (33) is significantly moreefficient computationally that the original triple integral of (15).

Some limit cases are interesting to solve analytically. For example, ifthe phase contribution (υ_(K,k)−ω_(K,k))=0 and the reflectivities R, Zare independent of incident angle and wavelength, then Eq. (33)simplifies to

$\begin{matrix}{q_{K,h} = {{{\delta_{K}\left( {R + Z} \right)}{\int_{0}^{\infty}{\int_{{\hat{\kappa}/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}\ {\mathbb{d}k}\ {\mathbb{d}\hat{\kappa}}}}}} + {H_{K}{\exp\left( {{\mathbb{i}}\; K\; h} \right)}\sqrt{R\; Z}{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{K,k}{\mathbb{d}k}}}} + {H_{- K}{\exp\left( {{\mathbb{i}}\; K\; h} \right)}\sqrt{R\; Z}{\int_{{{- K}/2}n_{0}}^{\infty}{\Gamma_{{- K},k}{\mathbb{d}k}}}}}} & (34)\end{matrix}$and we have only to handle integrals involving the weighting factorΓ_(K,k) defined in Eq. (24). This idealized case simplifies evaluationof two further limit cases for Eq. (34): Near-monochromatic illuminationwith a high-NA objective, and broadband illumination with low-NA.

For the case of a near-monochromatic light source having a narrowspectral bandwidth k_(Δ), we have the normalized spectrum

$\begin{matrix}{V_{k} = {\frac{1}{k_{\Delta}}H_{({k - {k\; 0}})}H_{({{k\; 0} + k_{\Delta} - k})}}} & (35)\end{matrix}$where k0 is the nominal source wavenumber. The integrations in Eq. (34)are now of the form:

$\begin{matrix}{{\int_{0}^{\infty}{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}\ {\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}} = {\frac{1}{4n_{0}^{2}k_{\Delta}}{\int_{0}^{\infty}{H_{({{k\; 0} - {{\hat{\kappa}/2}n_{0}}})}\hat{\kappa}{\int_{k\; 0}^{{k\; 0} + k_{\Delta}}{\frac{U_{\hat{\kappa},k}}{k^{2}}\ {\mathbb{d}k}\ {\mathbb{d}\hat{\kappa}}}}}}}} & (36) \\{{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{K,k}\ {\mathbb{d}k}}} = {\frac{1}{4n_{0}^{2}k_{\Delta}}H_{({{k\; 0} - {{K/2}n_{0}}})}K{\int_{k\; 0}^{{k\; 0} + k_{\Delta}}{\frac{U_{K,k}}{k^{2}}\ {{\mathbb{d}k}.}}}}} & (37)\end{matrix}$Assuming that U_(K,k) is essentially constant over the small bandwidthk_(Δ), we have

$\begin{matrix}{{\int_{0}^{\infty}{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}\ {\mathbb{d}k}\ {\mathbb{d}\hat{\kappa}}}}} = {\int_{0}^{\infty}{H_{({{k\; 0} - {{\hat{\kappa}/2}n_{0}}})}U_{\hat{\kappa},{k\; 0}}\frac{\hat{\kappa}}{4n_{0}^{2}k\; 0^{2}}\ {\mathbb{d}\hat{\kappa}}}}} & (38) \\{{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{K,k}\ {\mathbb{d}k}}} = {H_{({{k\; 0} - {{K/2}n_{0}}})}U_{K,{k\; 0}}{\frac{K}{4n_{0}^{2}k\; 0^{2}}.}}} & (39)\end{matrix}$where in the evaluation of the integrals we have used

$\begin{matrix}{{{\frac{- 1}{{k\; 0} + k_{\Delta}} + \frac{1}{k\; 0}} \approx \frac{k_{\Delta}}{k\; 0}},} & (40)\end{matrix}$valid for a narrow bandwidth k_(Δ)<<k0. In particular, the positive,nonzero portion of the spectrum reduces to

$\begin{matrix}{q_{{K > 0},h} = {\frac{H_{K}H_{({{k\; 0} - {{K/2}n_{0}}})}U_{K,{2n_{0}k\; 0}}K\sqrt{RZ}}{4n_{0}^{2}k\; 0^{2}}{\exp\left( {{\mathbb{i}}\;{Kh}} \right)}}} & (41)\end{matrix}$Consequently, for this special case of a narrow spectral bandwidth lightsource, constant reflectivities R, Z and no phase contributions ω,φ″_(K,h) =Kh.  (42)In this special case, the phase is linearly proportional to surfaceheight, consistent with conventional FDA. The spatial frequency also hasa direct correspondence to the directional cosine:K=β2n ₀ k0.  (43)Thus there is a one-to-one relationship between the spatial frequencycoordinate of the FDA spectra and the angle of incidence. Note furtherthe K weighting in the Fourier magnitude √{square root over (Q_(K))}calculated from Eq. (41). This is evident in the example spectrum FIG.6( a), which shows the theoretical prediction for a perfectly uniformfilling of the pupil plane over a range starting from normal incidenceup to the directional cosine limit imposed by the objective NA:β_(NA)=√{square root over (1−NA ²)}  (44)As a second example, consider the case of broadband illumination withuniform illumination restricted to a narrow range β_(Δ) of directionalcosines near normal incidence. The normalize pupil plane distribution isthen

$\begin{matrix}{U_{\beta} = {\frac{1}{\beta_{\Delta}}H_{1 - \beta}{H_{\beta - {({1 - \beta_{\Delta}})}}.}}} & (45)\end{matrix}$After the change of variables,

$\begin{matrix}{U_{K,k} = {\frac{1}{\beta_{\Delta}}H_{({{2{kn}_{0}} - K})}H_{\lbrack{K - {2{{kn}_{0}{({1 - \beta_{\;\Delta}})}}}}\rbrack}}} & (46)\end{matrix}$The definite integrals in Eq. (34) are in this case of the form

$\begin{matrix}{{\int_{0}^{\infty}{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{\hat{\kappa},k}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}} = {\frac{1}{\beta_{\Delta}}{\int_{0}^{\infty}{\int_{{\hat{\kappa}/2}n_{0}}^{{\hat{\kappa}/{({1 - \beta_{\Delta}})}}2n_{0}}{\frac{V_{k}\hat{\kappa}}{4k^{2}n_{0}^{2}}{\mathbb{d}k}{\mathbb{d}\hat{\kappa}}}}}}} & (47) \\{{\int_{{K/2}n_{0}}^{\infty}{\Gamma_{K,k}{\mathbb{d}k}}} = {\frac{1}{\beta_{\Delta}}{\int_{{K/2}n_{0}}^{{K/{({1 - \beta_{\Delta}})}}2n_{0}}{\frac{V_{k}K}{4k^{2}n_{0}^{2}}{\mathbb{d}k}}}}} & (48)\end{matrix}$which evaluate to

$\begin{matrix}{{\int_{0}^{\infty}{\int_{{K/2}\; n_{0}}^{\infty}{{\;\Gamma_{\hat{\kappa},\; k}}\;{\mathbb{d}k}\;{\mathbb{d}\hat{\kappa}}}}}\; = \;{\int_{0}^{\infty}{\frac{V_{{\hat{\kappa}/2}n_{0}}}{2\; n_{0}}\mspace{11mu}{\mathbb{d}\hat{\kappa}}}}} & (49) \\{{\int_{{K/2}\; n_{0}}^{\infty}{{\;\Gamma_{{K,\; k}\;}}\;{\mathbb{d}k}}}\; = {\frac{V_{{K/2}n_{0}}}{\; n_{0}}.}} & (50)\end{matrix}$where we have used

$\begin{matrix}{{\frac{\left( {1 - \beta_{\Delta}} \right)2n_{0}}{\hat{\kappa}} - \frac{2n_{0}}{\hat{\kappa}}} = {- \frac{2n_{0}\beta_{\Delta}}{\hat{\kappa}}}} & (51)\end{matrix}$The positive, nonzero portion of the spectrum is for this broadbandsource illumination and near-normal incidence is therefore

$\begin{matrix}{q_{{K > 0},h} = {\frac{V_{{K/2}n_{0}}\sqrt{RZ}}{2n_{0}}{\exp\left( {{\mathbb{i}}\;{Kh}} \right)}}} & (52)\end{matrix}$This corresponds closely to the familiar result that the Fouriermagnitude √{square root over (Q_(K))} is proportional to the sourcespectrum distribution V_(K/2n) ₀ , as shown e.g. in FIG. 6( b) for agaussian spectrum centered on a nominal or mean wavelength k0. Note thatEq. (52) also conforms to the assumption of linear phase evolutionφ″_(K,h) =Kh  (53)consistent with conventional FDA.

Since the Fourier magnitude √{square root over (Q_(K,h))}=|q_(K,h)| andphase φ″_(K,h)=arg(q_(K,h)) are derived from the Fourier Transform ofthe interference intensity I_(ζ,h), the inverse transform puts us backinto the domain of real interference signals

$\begin{matrix}{I_{\zeta,h} = {\int_{- \infty}^{\infty}{q_{\hat{\kappa},h}{\exp\left( {{- {\mathbb{i}}}\;\hat{\kappa}\zeta} \right)}{\mathbb{d}\hat{\kappa}}}}} & (54)\end{matrix}$where once again we have used {circumflex over (κ)} as for the spatialfrequency to emphasize that it is a free variable of integration in Eq.(54). Thus one way to calculate the intensity signal is to generate theFourier components q_(K,h) by Eq. (33) and transform to I_(ζ,h) usingEq. (54).

We assume random polarization of the source light in the present model.This does not mean, however, that we should neglect polarizationeffects. Rather, in the above calculations, we assume an incoherentsuperposition of equally weighted results from the two orthogonalpolarization states s and p defined by the plane of incidence of theillumination. Using superscript notation for the polarizations,q _(β,k) =q _(β,k) ^(s) +q _(β,k) ^(p).  (55)Therefore, the average phase angle for unpolarized light at this β, kwould be<φ″_(β,k)>=arg(q _(β,k) ^(s) +q _(β,k) ^(p)).  (56)Note that unless the magnitudes are identical for the two polarizationcontributions, most often<φ″_(β,k)>≠(φ″_(β,k) ^(s)+φ″_(β,k) ^(p))/2.  (57)Also, unless q_(β,k) ^(s) and q_(β,k) ^(p) are perfectly parallel in thecomplex plane,<Q _(β,k)>≠(Q _(β,k) ^(s) +Q _(β,k) ^(p))/2.  (58)The same observation applies to the system and object reflectivitiesR_(β,k) ^(s), R_(β,k) ^(p) and Z_(β, k) ^(s), Z_(β,k) ^(p),respectively; they cannot be summed directly unless they have identicalphases.

Provided that we take proper care of the polarization effects in thecalculation of the object surface reflectivity, the modeling remainsfairly straightforward and is flexible enough to handle the moreinteresting cases of polarized light further down the line.

The next step is to translate to discrete numerical formulas, in view ofa software development. We redefine the relationship between theinterference signal I_(ζ,h) and the Fourier spectrum q_(K,h) usingdiscrete Fourier transforms as

$\begin{matrix}{q_{K,h} = {\frac{1}{\sqrt{N}}{\sum\limits_{\zeta}^{\;}{I_{\zeta,h}{\exp\left( {{\mathbb{i}}\; K\;\zeta} \right)}}}}} & (59) \\{I_{\zeta,h} = {\frac{1}{\sqrt{N}}\left\lbrack {q_{0} + {\sum\limits_{K > 0}^{\;}{q_{K,h}{\exp\left( {{- {\mathbb{i}}}\; K\;\zeta} \right)}}} + {\sum\limits_{K > 0}^{\;}{{\overset{\_}{q}}_{K,h}{\exp\left( {{\mathbb{i}}\; K\;\zeta} \right)}}}} \right\rbrack}} & (60)\end{matrix}$where q _({circumflex over (κ)},h) is the complex conjugate ofq_({circumflex over (κ)},h) and there are N discrete samples in theinterference signal I_(ζ,h). In Eq. (60) and what follows, we have setaside the use of a free variable {circumflex over (κ)} that wasimportant in the derivations but it is no longer needed as a substitutefor the spatial frequency K. The predicted positive-frequency FDAcomplex spectrum is then

$\begin{matrix}{q_{{K \geq 0},h} = {\rho_{K \geq 0}{\exp\left( {{\mathbb{i}}\;{Kh}} \right)}}} & (61)\end{matrix}$where the normalized, height-independent coefficients are

$\begin{matrix}{\rho_{K > 0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{k}^{\;}{H_{k - {{K/2}n_{0}}}\sqrt{R_{K,k}Z_{K,k}}{\exp\left\lbrack {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{K,k}} \right)} \right\rbrack}\Gamma_{K,k}}}}} & (62) \\{\rho_{0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{K \geq 0}^{\;}{\sum\limits_{k}^{\;}{\left( {R_{K,k} + Z_{K,k}} \right)\Gamma_{K,k}}}}}} & (63)\end{matrix}$where the normalization for the range of integration is

$\begin{matrix}{\Upsilon = {\sum\limits_{K \geq 0}{\sum\limits_{k}{H_{k - {{K/2}n_{0}}}\Gamma_{\;{K,\; k}}}}}} & (64)\end{matrix}$The Heaviside step functions H in Eq. (62) prevent unnecessarycontributions to the sums. The weighting factor Γ_(K,k) is as defined inEq. (24).

To compare experiment with theory, we use Eq. (61) to generate anexperimental FDA spectrum and Eq. (62) to transform back into the spacedomain for the theoretical prediction of I_(ζ,h). This is mostefficiently performed by fast Fourier transforms (FFT). The propertiesof the FFT determine the range of K values. If the N discrete samplesfor I_(ζ,h) are spaced by an increment ζ_(step), there will be N/2+1positive spatial frequencies starting from zero and rising to N/2 cyclesper data trace, spaced by an increment

$\begin{matrix}{K_{step} = {\frac{2\pi}{N\;\zeta_{step}}.}} & (65)\end{matrix}$To facilitate phase unwrapping in the frequency domain, we try to adjustthe zero position for the scan so that it is near the signal peak, thusreducing the phase slope in the frequency domain. Because an FFT alwaysassumes that the first data point in the scan is at zero, the signalshould be properly offset.

We now focusing on modeling a sample surface with a thin film.

FIG. 7 shows two surface types, with and without a thin film. For bothcases, we define an effective amplitude reflectivity z_(β,k) accordingtoz _(β,k)=√{square root over (Z_(β,k))}exp(iω _(β,k))  (66)where Z_(β,k) is the intensity reflectivity and ω_(β,k) is the phasechange on reflection. The subscripts β,k emphasize a dependency on thedirectional cosine of the illuminationβ₀=cos(ψ₀),  (67)where ψ₀ is the incident angle, and on the wavenumberk=(2π/λ).  (68)where λ is the wavelength of the light source. The subscript β will beunderstood to refer to the first incident directional cosine β₀.

The surfaces are characterized in part by their index of refraction. Theindex of the surrounding medium, usually air, is n₀. For the simplesurface FIG. 7( a) there is only one index n₁. For the thin film in FIG.7( b), there are two surface indices, n₁ for the transparent orpartially transparent film and n₂ for the substrate. Most generally,these refractive indices are complex numbers characterized by a realpart and an imaginary part. For example, a typical index, e.g. forchrome at λ=550 nm, is n₁=3.18+4.41i, where we are adopting theconvention where the imaginary part is defined as positive.

The index of refraction of a material depends on the wavelength. Thedispersion in refractive index n₀ for the air is not very significant,but is important for many sample surfaces, especially metals. Over smallwavelength changes near a nominal k0, most materials have a nearlylinear dependence on wavenumber, so that we can write

$\begin{matrix}\begin{matrix}\; \\{n_{1,k} = {v_{1}^{(0)} + {kv}_{1}^{(1)}}} \\\;\end{matrix} & (69)\end{matrix}$where v₁ ⁽⁰⁾,v₁ ⁽¹⁾ are the intercept and slope, respectively, for theindex of refraction n₁ at the nominal wavenumber k0.

The most common use of the refractive index is Snell's law. Referring toFIG. 7( b), the refracted beam angle internal to the film is

$\begin{matrix}{\begin{matrix}\; \\{\psi_{1,\beta,k} = {\arcsin\left\lbrack {\frac{n_{0}}{n_{1,\beta,k}}{\sin\left( \psi_{0} \right)}} \right\rbrack}} \\\;\end{matrix},} & (70)\end{matrix}$where ψ₀ is the angle within the medium of index n₀ incident on the topsurface of the medium of index n₁, and ψ_(1,β,k) is the angle ofrefraction. It is possible for these angles to take on complex values ifthe indices are complex, indicating partially evanescent propagation.

The complex amplitude reflectivity of a boundary between two mediadepends on the polarization, the wavelength, the angle of incidence andthe index of refraction. The s- and p-polarization reflectivities of thetop surface of the film in FIG. 7( b) are given by the Fresnel formulae

$\begin{matrix}\begin{matrix}\; \\{\vartheta_{1,\beta,k}^{s} = \frac{\tan\left( {\psi_{0} - \psi_{1,\beta,k}} \right)}{\tan\left( {\psi_{0} + \psi_{1,\beta,k}} \right)}} \\\;\end{matrix} & (71) \\\begin{matrix}\; \\{\vartheta_{1,\beta,k}^{p} = {- \frac{\sin\left( {\psi_{0} - \psi_{1,\beta,k}} \right)}{\sin\left( {\psi_{0} + \psi_{1,\beta,k}} \right)}}} \\\;\end{matrix} & (72)\end{matrix}$The dependence on β,k results from the angles ψ₀,ψ_(1,β,k), the exitangle ψ_(1,β,k) introducing a k dependency via the refractive indexn_(1,k). Similarly, the substrate-film interface reflectivities are

$\begin{matrix}{\vartheta_{2,\beta,k}^{p} = \frac{\tan\left( {\psi_{1,\beta,k} - \psi_{2,\beta,k}} \right)}{\tan\left( {\psi_{1,\beta,k} + \psi_{2,\beta,k}} \right)}} & (73) \\{\vartheta_{2,\beta,k}^{s} = {- \frac{\sin\left( {\psi_{1,\beta,k} - \psi_{2,\beta,k}} \right)}{\sin\left( {\psi_{1,\beta,k} + \psi_{2,\beta,k}} \right)}}} & (74)\end{matrix}$Note that in the Fresnel equations, if the angle of incidence andrefraction are the same, the reflectivity for both polarizations goes tozero.

For a simple surface (no thin film), the sample surface reflectivity isidentical to the top-surface reflectivityz _(β,k)=φ_(1,β,k) (simple surface, no thin film)  (75)Consequently, the phase change on reflection (PCOR) caused by thesurface reflection isω_(β,k)=arg(φ_(1,β,k)),  (76)Note that to satisfy the boundary conditions, the s-polarization “flips”upon reflection (=π phase shift for a dielectric) whereas thep-polarization does not. The distinction between polarization statesbecomes meaningless exactly at normal incidence, which in any caseresults in a division by zero in the Fresnel equations and a differentformula handles this limit case.

When using the plus sign convention for the complex part of the index ofrefraction, the greater the absorption (complex part), the greater thePCOR ω_(β,k). In other words, a larger absorption coefficient isequivalent to a decrease in effective surface height. This makesintuitive sense—one imagines absorption as a penetration of the lightbeam into the material prior to reflection, rather than a cleanreflection and transmission right at the boundary. Following our usualconvention, for which an increase in surface height corresponds to apositive change in the phase difference between the reference andmeasurement surfaces, a positive surface PCOR subtracts from theinterferometer phase.

A thin film is a special case of a parallel plate reflection. The lightpasses through the top surface partially reflected (see FIG. 7) andcontinues to the substrate surface where there is a second reflectionwith a phase delay with respect to the first. However, this is not theend of the story. The light reflected from the substrate is once againpartially reflected when passing back up through the top surface,resulting in an additional reflected beam heading south again to thesubstrate. This goes on in principle forever, with each additionalreflection a little weaker than the last. Assuming that all of thesemultiple reflections survive to contribute to the final surfacereflectivity, the infinite series evaluates to

$\begin{matrix}\begin{matrix}\; \\{z_{\beta,k} = \frac{\vartheta_{1,\beta,k} + {\vartheta_{2,\beta,k}{\exp\left( {2{\mathbb{i}}\;{kL}\;\beta_{1,\beta,k}n_{1,k}} \right)}}}{1 + {\vartheta_{1,\beta,k}\vartheta_{2,\beta,k}{\exp\left( {2{\mathbb{i}}\;{kL}\;\beta_{1,\beta,k}n_{1,k}} \right)}}}} \\\;\end{matrix} & (77)\end{matrix}$β_(1,β,k)=cos(ψ_(1,β,k)).  (78)

As a note of clarification, recall the β dependency of β_(1,⊕,k) refersto a dependency on the incident directional cosine β₀ in the ambientmedium of index n₀. The same Eq. (77) applies to both polarizationstates, with corresponding single-surface reflectivities.

Inspection of these equations shows why conventional FDA processingbreaks down in the presence of thin films. Conventional FDA determinessurface height by a linear fit to the Fourier phase spectrum weighted bythe Fourier power spectrum, using broadband (white) light to generatethe Fourier spatial frequency spread. The idea is that the phaseevolution comes from the expected linear phase dependence on surfaceheight. Any other constant offset or linear coefficients (e.g.,“dispersion”) associated with the surface characteristics are removed bysystem characterization or by simply ignoring those phase contributionsthat do not change with field position.

This works perfectly fine for simple surfaces. With unpolarized light,and most likely with the circularly-polarized light, the wavelengthdependence of the PCOR is nearly linear with respect to wavenumber andconstant for a given material. In the presence of a thin film, however,the conventional analysis breaks down. The phase becomes nonlinear andthe phase slope becomes sensitive to film thickness, which may bevarying across the field of view. Therefore, the present analysisdetermines key parameters of the surface structure such as filmthickness by comparing experimental data to a theoretical prediction,using our knowledge of how e.g. a thin film modulates the reflectivityof the surface.

We now discuss how comparison of experimental data to a library oftheoretical predictions provides surface structure parameters such asfilm thickness and phase change on reflection (PCOR). In the case of afilm of unknown thickness, the library for a single surface type, e.g.SiO₂ on Si, would range over many possible film thicknesses. Infrequency domain embodiments, the idea is to search this library for amatch to those characteristics of the FDA spectra that are independentof surface topography, for example, a distinctive structure to themagnitude spectrum resulting from a thin-film interference effect. Thecomputer then uses the library spectrum to compensate the FDA data,allowing for a more accurate surface topography map.

In one embodiment, the library contains example FDA spectra for surfacestructures, each spectrum providing a series of complex coefficientsρ_(K) representing Fourier coefficients as a function of spatialfrequency K. These spectra are the Fourier transforms of intensity dataI_(ζ,h) acquired during a scan ζ of the optical path length of aninterferometer. specific description above often refer to a scanninginterferometry signal for which limited coherence in the interferometrysystem causes localization of the interference fringes; for manyembodiments, it is also possible to extract information about complexsurface features from interferometry signal(s) not having such fringelocalization.

For example, interferometry signal from different locations of the testobject that do not have fringe localization can still be used togenerate an apparent surface profile for the test object, and thatapparent surface profile, or information derived there from, can becompared to models of the expected response for different values oflateral surface features of the test object that are not-resolved orobscured in the apparent surface profile to determine information aboutsuch under-resolved features in much the same way as that describedabove for low coherence scanning interferometry signals. Techniques forextracting surface profile information from such “high ” coherenceinterferometry signals are generally referred to as phase shiftinginterferometry (PSI) algorithms, and are well-known in the art. See, forexample, the background and contents of U.S. Pat. No. 6,359,692,entitled “METHOD AND SYSTEM FOR PROFILING OBJECTS HAVING MULTIPLEREFLECTIVE SURFACES USING WAVELENGTH-TUNING PHASE-SHIFTINGINTERFEROMETRY,” the contents of which are incorporated herein byreference. To generate the interferometry data for such PSI analysis,the interferometry signal for a given pixel can be generated bymechanically varying the optical path length difference between thereference and measurement legs, or by varying the wavelength of thelight for a fixed, non-zero optical path length difference between thereference and measurement legs.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention. Thespatial frequency K is proportional to the angular wavenumber k=2π/λ fora segment of the source light spectrum, the index of refraction n₀ ofthe ambient medium, and the directional cosine β=cos(ψ), where ψ is theangle of incidence for a ray bundle directed to the object surface:K=2βkn₀.  (79)The ρ_(K) coefficients for the prediction library include the opticalproperties of the surface that can influence the appearance of the FDAspectra, with the exception of surface height.

Predicting the FDA spectra involves an integral representing theincoherent sum of ray bundles over a range of incident angles ψ andangular wavenumbers k for the source light. As described above, thenumerical integration can reduce to a computationally-efficient singlesum over N angular wavenumbers k, weighted by a factor Γ_(K,k):

$\begin{matrix}{\rho_{K > 0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{k}{H_{k - {{K/2}n_{0}}}\sqrt{R_{K,k}Z_{K,k}}{\exp\left( {{\mathbb{i}}\left( {\upsilon_{K,k} - \omega_{K,k}} \right)} \right\rbrack}\Gamma_{K,k}}}}} & (80) \\{\rho_{0} = {\frac{\sqrt{N}}{\Upsilon}{\sum\limits_{K \geq 0}{\sum\limits_{k}{{H_{k - {{K/2}n_{0}}}\left( {R_{K,k} + Z_{K,k}} \right)}\Gamma_{K,k}}}}}} & (81)\end{matrix}$The weighting factor is

$\begin{matrix}{{\Gamma_{K,k} = \frac{{KU}_{K,k}V_{k}}{4k^{2}n_{0}^{2}}},} & (82)\end{matrix}$where V_(k) is the source spectrum and U_(K,k) is the pupil-plane lightdistribution. The corresponding normalization Υ is the sum over allspatial frequencies of the weighting factor

$\begin{matrix}{\Upsilon = {\sum\limits_{K \geq 0}{\sum\limits_{k}{H_{k - {{K/2}n_{0}}}{\Gamma_{K,k}.}}}}} & (83)\end{matrix}$where Υ is a normalization to be defined shortly and H is the Heavisidestep function.

The distinctive characteristics of an object surface structure,particularly of a thin film, enter into the spectrum ρ_(K) through theobject-path phase ω_(K,k) and reflectivity Z_(K,k), as detailed above.Equally important are the reference-path phase υ_(K,k) and reflectivityR_(K,k), which depend on the scanning interferometer itself. Suchfactors can be determined by theoretically modeling the scanninginterferometer or by calibrating it with a test sample having knownproperties, as described further below.

The typical prediction library for a thin film is a series of spectraρ_(K) indexed by film thickness L. The stored spectra cover only anarrow spatial frequency region of interest (ROI), usually 15 or 16values for a 256-frame intensity data acquisition, the remainder of thevalues outside this ROI being zero. The limits of the ROI follow fromthe definition of the spatial frequency:K^(min)=2β^(min)k^(min)n₀  (84)K^(max)=2β^(max)k^(max)n₀  (85)A typical range of spatial frequencies for a scanning interferometerbased on a 100× Mirau objective and a narrow bandwidth, 500-nm lightsource is 2.7 μm⁻¹ to 4.0 μm⁻¹. For computational efficient, a denselook up table, indexed by 0.5 to 5 nm between sample spectra, can beused rather than an analytical search routine that involvesrecalculation using Eqs. (80)-(83) several times for each pixel.

The library search involves the following steps: (1) Select a predictedFDA spectrum from the library corresponding to a specific surface type,(2) calculate how closely this spectrum matches the experimental datausing a merit function, then (3) repeat through several or all of thelibrary data sets to determine which theoretical spectrum provides thebest match. What we are looking for is a “signature” in the frequencydomain that relates uniquely to surface characteristics such as thinfilms, dissimilar materials, step structures, roughness, and theirinteraction with the optical system of the interferometer. Thiscomparison therefore explicitly filters away the linear rate of changeof phase with spatial frequency, which is the one characteristic of theFDA spectrum that varies directly with surface topography and istherefore irrelevant to the library search.

In comparing spectra, there is a benefit to separating the phase andmagnitude contributions to the merit calculation. Thus for the theory,we haveP _(K)=|ρ_(K)|  (86)φ_(K)=connect_(K)[arg(ρ_(K))],  (87)where connect_(K) is a function that removes 2-π steps in the spatialfrequency dependence of φ_(K,h). For the experimental data we haveP _(K) ^(ex) =|q _(K,h) ^(ex)|  (88)φ″_(K,h) ^(ex)=connect_(K)[arg(q _(K,h) ^(ex))],  (89)The double prime for φ″_(K) ^(ex) indicates an uncertainty in the fringeorder from both pixel to pixel and overall with respect to the startingpoint in the scan. The experimental data necessarily include a slopeterm related to the local surface height; this is the reason for the useof the q symbol instead of the ρ symbol.

For a specific set of trial surface parameters, we can calculate a phasedifference

$\begin{matrix}\begin{matrix}\; \\{Ϛ_{K,h}^{''} = {\phi_{K,h}^{''_{ex}} - \phi_{K}}} \\\;\end{matrix} & (90)\end{matrix}$The phase difference ξ″_(K,h) is the compensated FDA phase, assumingthat the trial parameters are correct. A good match of theory toexperiment yields a phase ξ″_(K,h) that in principle is a simple linearfunction of spatial frequency K with an intercept of zero (i.e., zerophase gap). Thus, looking ahead, the successfully compensated phaseξζ″_(K,h) is what we shall eventually feed downstream to a conventionalFDA analysis, which assumes that the slope of the phase in frequencyspace is directly proportional to surface height.

Based on the observations of the previous paragraph, there are twofeatures of interest in the compensated phase ξ″_(K,h) that allow us toevaluate the match of theory to experiment independent of surfaceheight. The first is the phase gap A″ or K=0 intercept value ξ″_(K=0,h)obtained by a linear fit, and the second is the residual nonlinearitywith respect to wavenumber after a linear fit. Corresponding meritfunctions are, for example,

$\begin{matrix}{\chi_{0} = \left\lbrack {\frac{A^{''}}{2\pi} - {{round}\mspace{11mu}\left( \frac{A^{''}}{2\pi} \right)}} \right\rbrack^{2}} & (91) \\{\chi_{\phi\;{non}} = \frac{\sum\limits_{K > 0}{\left( {ϛ_{K,h}^{''} - {\sigma_{h}K} - A^{''}} \right)P_{K,h}^{ex}}}{\sum\limits_{K > 0}P_{K,h}^{ex}}} & (92)\end{matrix}$where σ_(h) is the slope of the (magnitude weighted) linear fit to thecompensated phase ξ″_(K,h). The round ( ) function in Eq. (91) limitsthe phase gap A″ to the range ±π.

Although a library search can proceed using phase information alone,i.e. by minimizing one or both of the merit function values χ_(φ) and/orχ_(φnon), we also have important and useful signatures in the Fouriermagnitude. The magnitude is particularly interesting in that it isinherently independent of surface height. Thus for example, we candefine in approximate analogy with the phase merits the followingmagnitude merit functions:

$\begin{matrix}{\chi_{P} = \left\lbrack \frac{\sum\limits_{K > 0}\left( {P_{K,h}^{ex} - P_{K,h}} \right)}{\sum\limits_{K > 0}\left( {P_{K,h}^{ex} + P_{K,h}} \right)} \right\rbrack^{2}} & (93) \\{\chi_{P\;{non}} = \frac{\sum\limits_{K > 0}\left( {{\Omega^{- 1}P_{K,h}^{ex}} - P_{K,h}} \right)^{2}}{\sum\limits_{K > 0}\left( {{\Omega^{- 1}P_{K,h}^{ex}} + P_{K,h}} \right)^{2}}} & (94)\end{matrix}$where Ω is the empirical scaling factor

$\begin{matrix}{\Omega = {\sum\limits_{K > 0}{P_{K,h}^{ex}/{\sum\limits_{K > 0}{P_{K,h}.}}}}} & (95)\end{matrix}$The merit χ_(P) is most closely related to the overall reflectivity ofthe object surface, independent of spatial-frequency dependence, whereasχ_(Pnon) expresses how well the theoretical and experimental magnitudeplots match in shape.

The magnitude merit functions χ_(P) and/or χ_(Pnon) are in addition toor even in place of the phase merits χ_(φ) and/or χ_(φnon). A generallibrary search merit function is therefore

$\begin{matrix}\begin{matrix}\; \\{\chi = {{w_{\phi}\chi_{\phi}} + {w_{\phi\;{non}}\chi_{\phi\;{non}}} + {w_{P}\chi_{P}} + {w_{P\;{non}}\chi_{P\;{non}}}}} \\\;\end{matrix} & (96)\end{matrix}$where the w are weighting factors. In principle, one can determine theweights in Eq. (96) knowing the standard deviation for the variousparameters. A more empirical approach is to try out various weights onreal and simulated data and see how well they work. For the examplesthat follow, we select equal weights w_(φ)=w_(φnon)=w_(P)=w_(Pnon)=1 forall merit contributions.

The examples in FIGS. 8-13 illustrate the merit-function searchprocedure for six SiO₂ on Si film thicknesses: 0, 50, 100, 300, 600, and1200 nm, respectively. A single library for all examples encompasses therange from 0 to 1500 nm in 2-nm intervals. The data are simulations,free of noise. As in all the examples described herein, the scan step is40 nm, the source wavelength is 498 nm, and the source gaussian FWHM is30 nm (quasi-monochromatic).

The most interesting aspect of these simulated searches is the behaviorof the four merit functions. Generally, we observe that inclusion ofthese four functions helps to reduce the ambiguity in the final meritvalues, there being a strong periodicity for individual merit values asa function of film thickness. Another general observation is that themerits based on nonlinearity, both in phase and magnitude, are mosteffective at 300 nm and above, whereas the phase gap and averagemagnitude are dominant below 300 nm film thickness. This shows that theχ_(φ),χ_(P) merit functions are especially useful to the really thinfilms, which places importance on system characterization, which couplesdirectly into the phase gap and magnitude results.

Once we determine the thin film thickness (or identify the material orother uses for the algorithm), FDA processing proceeds in the usual way,using however the corrected FDA phase ζ″_(K,h) instead of the originalexperimental phase data. In principle, if the modeling has beensuccessful, ζ″_(K,h) should be free of nonlinearities and the phase gapshould be zero. The next step therefore is a linear fit to the phasespectrum ζ″_(K,h). It appears more effective for high-NA FDA to use themagnitude spectrum P_(K) in place of magnitude squared. The fit providesfor each pixel a slopeσ_(h) ≈dζ″ _(K,h) /dK  (97)and an intercept (phase gap)A″≈ζ″ _(K=0,h).  (98)Note that the phase gap A″ carries the double prime inherited from thefringe order uncertainty in the phase data. The slope σ_(h) is free ofthis uncertainty. From the intercept A″ and the slope σ_(h), we definefor a specific mean or nominal spatial frequency K0 a “coherenceprofile”Θ_(h)=σ_(h) K0  (99)and a “phase profile”θ″_(h)=Θ_(h) +A″.  (100)We then removes the pixel-to-pixel fringe order uncertainty in the phaseθ″_(h):

$\begin{matrix}{\theta^{\prime} = {\theta^{''} - {2\pi\mspace{14mu}{{round}\mspace{11mu}\left\lbrack \frac{A^{''} - \alpha^{\prime}}{2\pi} \right\rbrack}}}} & (101)\end{matrix}$where α′ is an approximation to the original phase gap A″ that is freeof pixel-to-pixel 2π steps.

Finally, the height profile follows fromh′=θ′/K0.  (102)Note that it is not necessary to subtract the phase offset γ, because ithas already been done in generating the compensated phases ζ_(K,h).

The first example of a surface topography measurement (FIG. 14) is apure simulation. The surface topography is everywhere zero, but there isan underlying film layer that progresses from 0 to 1500 nm in 10 nmincrements. Using the same prediction library as in FIGS. 8-13, thistest demonstrates unambiguous determination of film thickness throughoutthe range of the prediction library, albeit for perfect, noise-freedata.

The next example (FIG. 15) is also a simulation, but with additivenoise. The random additive noise is gaussian, with a standard deviationof 2 bits out of an average 128 intensity bits, which looks to betypical of real data. The results are clearly satisfactory despite thesignificant difference in reflectivity between SiO₂ and Si (4% to 45%).

We now address system characterization.

We define a phase offset γ_(sys) and a linear dispersion τ_(sys) usingdata collected during a system characterization procedure. To includesystem characterization data, we correct the Fourier-transformedexperimental data q_(K) ^(ex) prior to the library search and prior toany other FDA processing on a pixel-by-pixel basis usingq _(K>0) ^(ex) =M ⁻¹exp[−iγ _(sys) −i(K−K0)τ_(sys) ]q _(K>0)^(ex).  (103)where K0 is the nominal spatial frequency, which represents the nominalspectral frequency for the FDA data set, as identified e.g. by locatingthe midpoint of the ROI. Note that the theoretical library remainsunchanged. The scaling coefficient M (greek capital “M”) is a new systemcharacterization that makes it possible to use object surfacereflectivity as a parameter in the library search.

The phase offset γ_(sys) and the system phase gap A_(sys) as a functionsof field position can be stored as a function of field position, andcalculate the true system dispersion according toτ_(sys)=(γ_(sys) −A _(sys))/K0.  (104)The magnitude coefficient M is also field dependent.

The creation of system characterization data proceeds in a mannersimilar to that described above for the object sample. We move to anartifact having known characteristics, measure it, and determine thesystem characterization by looking at how the results differ from whatwe would expect for a perfect system. Specifically, using a known samplefor which the correct library entry is predetermined, we generate thephase gap A″ as in Eq. (98) and a final height h′ as in Eq. (102). Then,assuming a perfectly flat artifact, we calculate the system phase offsetγ_(sys) =K0h′  (105)and the system phase gapA _(sys)=connect_(xy)(A″)  (106)where connect_(xy)( ) is pixel-to-pixel phase unwrapping. The magnitudemap is

$\begin{matrix}{M_{sys} = {\sum\limits_{K > 0}{P_{K,h}^{ex}/{\sum\limits_{K > 0}{P_{K,h}.}}}}} & (107)\end{matrix}$In some embodiments, several system characterizations can be averaged,perhaps using artifacts having similar surface structure to the finalapplication (e.g. SiO2 on Si) over a range of sample types.

In much of the description and simulations above we have focused on thinfilm surface structures, however, the analysis is also applicable toother types of complex surface structures. In what follows we show howthe scanning interferometry data can be analyzed to account for surfacestructures that are smaller than the optical resolution of the scanninginterferometer microscope. The optical resolution is ultimately limitedby the wavelength of the light source and the NA of the light collectionoptics.

FIG. 16 a shows height profiles determined from actual scanninginterferometry data of a 2400 lines per mm (1 pmm) grating having apeak-to-valley (PV) modulation depth of 120 nm using a light source at a500-nm nominal wavelength. The top profile in FIG. 16 a shows the heightprofile determined using a conventional FDA analysis. The conventionalanalysis indicates a PV modulation depth of only about 10 nm, greatlyunderestimating the actual modulation depth. This inaccuracy occursbecause the grating has features at the limit of the optical resolutionof the 500-nm instrument. This is so even though the pixel resolution ofthe camera in the instrument is more than sufficient to accuratelyresolve the grating.

One way of thinking about this effect is that the scanninginterferometry signal for a first camera pixel generally correspondingto a first surface location also includes contributions from adjacentsurface locations when those additional surface locations have surfacefeatures sufficiently sharp relative to the light wavelength to diffractlight to the first pixel. The surface height features from thoseadjacent surface locations corrupt conventional analysis of the scanninginterferometry signal corresponding to the first surface location.

At the same time, however, this means that the scanning interferometrysignal corresponding to the first surface location includes informationabout the complex surface features nearby. FIG. 17 illustrates this byshowing the scanning interferometry signal from pixels corresponding tovarious locations about a step height feature. For the signal in (a) thestep height is to the right of the pixel and higher, for the signal in(b) the step passes directly through the pixel, and for the signal in(c) the step is to the left of the pixel and is lower. One signaturethat is immediately apparent in the signals is the reduction in fringecontrast in (b) relative to (a) and (c). For example, if the step heightwas equal to one-quarter of the wavelength and the pixel locationcorresponded exactly to the position of the step height, the fringecontrast in (b) should disappear entirely because interference from thetwo sides of the step would exactly cancel one another. There is alsomuch information in the signals in shown in (a) and (c). For example,FIG. 18 shows the nonlinear distortions in the frequency domain phasespectra for the signals (a) and (c) of FIG. 17, respectively, resultingfrom the nearby step height. These spectra are indicated as (a) and (b),respectively, in FIG. 18. In the absence of the step height, thefrequency domain phase spectra would be linear. Thus, the nonlinearfeatures in the frequency domain phase spectrum for pixels correspondingto surface locations adjacent to the step height nonetheless includeinformation about the step height.

To more accurately measure the surface profile of a test surface in thepresence of such under-resolved surface features, we can use the librarysearching technique described above for thin films. For example, for thecase of a test surface with an under-resolved grating, a series of modelFDA spectra are generated for different values of the PV modulationdepth and offset position. As in the thin film examples, the surfaceheight for the model spectra remains fixed. The analysis then continuesas in the thin film examples above, except that rather than the modelspectra being parameterized by thin film thickness, they areparameterized by modulation depth and offset position. Comparisonbetween signatures of the FDA spectra for the actual test surface andthe different model spectra can then be used to determine a match. Basedon the match, distortions in the actual FDA spectrum for each pixelcaused by the presence of the grating are removed so that the surfaceheight for each pixel can be determined using conventional processing.The results of such an analysis using the same merit functions asdescribed above for the thin films are shown in FIGS. 16 b and 19 b.

FIG. 16 b shows the height profile determined using the library searchanalysis for 2400 lines per mm grating described above with reference toFIG. 16 a. The same data was used in the FIGS. 16 a and 16 b, however,the library search analysis determined the PV modulation depth for thegrating to be 100 nm, much closer to the actual 120-nm modulation depththan the 10-nm result determined by conventional FDA processing in FIG.16 a. FIGS. 19 a and 19 b show a similar analysis for a simulation witha discrete step height and assuming a nominal 500-nm light source. FIG.19 a shows the height profile determined using conventional FDAprocessing (solid line) compared to the actual height profile for thesimulation (dotted line). FIG. 19 b shows the height profile determinedusing the library search method (solid line) compared to the actualheight profile for the simulation (dotted line). The parameters for themodel spectra in the library search were location and step heightmagnitude. As illustrated, the library search analysis improves lateralresolution from about 0.5 microns to about 0.3 microns.

In the example of the FIGS. 19 a-b, the library was generatedtheoretically using an equation similar to Eq. (77) for a thin film,except in this case the denominator is set to 1 (because there are nomultiple reflections from underlying layers), the first and second termsin the numerator are weighted according to the lateral distance of themeasurement point from the actual step height position, and theparameter L corresponds to the step height itself rather than thin filmthickness. Thus this theoretical model is based on a complex summing ofrays emerging from either side of the step height. As the lateralposition of the pixel under inspection increases in distance from theposition of the step height, the signal tends to that of a simple flatsurface.

In the example of FIGS. 16 a-b, the library was generated experimentallyby observing the signal produced by the interferometry instrument forthe 2400 line grating. Based on this experimental data, signatures inthe scanning interferometry data were correlated with a correspondinglateral position in the period of the grating to build up the library.Experimental data for each pixel of the test sample (which in this casewas the same 2400 line grating) was then compared with the library todetermine the best lateral position in the period for that pixel.

In the detailed analyses described above the comparison betweeninformation in the actual data and information corresponding to thedifferent models has occurred in the frequency domain. In otherembodiments, the comparison can be made in the scan coordinate domain.For example, while changes in the absolute position of the fringecontrast envelope is generally indicative of changes in surface heightat a first surface location corresponding to the signal in question, theshape of the signal (independent of its absolute position) containsinformation of complex surface structure, such as underlying layers atthe first surface location and/or surface structure at adjacentlocations.

One simple case is to consider to the magnitude of the fringe contrastenvelope itself. For example, when a thin film thickness is very smallrelative to the range of wavelengths produced by the light source, theinterference effects produced by the thin film become wavelengthindependent, in which case thin film thickness directly modulates themagnitude of the fringe contrast envelope. So, in general, the fringecontrast magnitude can be compared to that for models corresponding todifferent thin film thicknesses to a identify a match for a particularthin film thickness (taking into account systematic contributions fromthe interferometer itself)

Another simple case is to look at the relative spacings of the zerocrossings of the fringes under the fringe contrast envelope. For asimple surface structure illuminated with a a symmetric frequencydistribution, the relative spacings between the different zero crossingsshould be nominally the same. Variations in the relative spacings aretherefore indicative of complex surface structure (when taking intoaccount systematic contributions from the interferometer itself) and cancompared to models for different complex surface structures to identifya match to a particular surface structure.

Another case is to perform a correlation between the scan-domain signaland the scan-domain signals corresponding to different models of thetest surface. A match generally corresponds to the correlation that hasthe highest peak value, which indicate the model whose scan-domainsignal has a shape most similar to the shape of the actual signal. Notethat such analysis is generally independent of surface height because adifference between the surface height of the actual sample and that ofeach model only shifts the location of peak in the correlation function,but does not effect, in genera) the peak value itself. On the otherhand, once the correct model is identified, the location of the peak inthe correlation function of the correct model yields the surface heightfor the test sample, without the need for further analysis (such asconventional FDA).

Like the analysis in the spatial frequency domain, an analysis in thescan-coordinate domain can be used for many different types of complexsurfaces, including not only thin films, but also other complex surfacestructures such as under-resolved surface height features as describedabove.

We now describe in detail a scan-coordinate library search analysis theinvolves a correlation between the signal for the test sample andcorresponding signals for various models of the test sample.

The approach sets aside any assumptions about the interference patternother than to say that all pixels in a data set corresponding to surfacelocations with the same complex surface characteristics contain the samebasic, localized interference pattern, only shifted in position (andpossibly rescaled) for each pixel. It does not matter what the signalactually looks like, whether it is a gaussian envelope or has a linearphase behavior in the frequency domain or whatever. The idea is togenerate a sample signal or template that represents this localizedinterference pattern for different models of complex surface structuresfor the test object, and then for each pixel, find the model whoselocalized interference pattern best matches the shape of the actuallocalized interference pattern, and for that model, find the scanposition within the data set that provides the best match between theinterference pattern template and the observed signal—which gives thesurface height. Several techniques are available for pattern matching.One approach is to mathematically correlate each template with the data.Using a complex (i.e. real plus imaginary) template function for eachmodel, we recover two profiles, one closely associated with the envelopeof the signal and the other associated with the phase of the underlyingcarrier signal.

In one embodiment, for example, the analysis for each pixel would beinclude: (1) selecting a test template from a library of templatescalculated or recorded for a specific value of an adjustable parameter,such as film thickness; (2) finding the local surface height using theselected test template and a correlation technique (an example of whichis described below); (3) recording the peak merit function value for theselected test template based on the correlation technique; (4) repeatingsteps 1-3 for all or a subset of the templates in the library; (5)determining which test template provides the best match (=highest peakmerit function value); (6) recording the value for the adjustableparameter for the best-matched template (e.g., thin film thickness); and(7) recalling the height value that provided the peak match positionwithin the data trace.

We now describe a suitable correlation technique based on a complexcorrelation. For each model of the test surface we generate a templateinterference patternI _(temp) ^(j)(ζ)=m _(temp) ^(j)(ζ)cos[K _(o)ζ+φ_(temp) ^(j)(ζ)]  (108),where the index j indicates the specific model for the template pattern.The functions m_(temp) ^(j)(ζ) and φ_(temp) ^(j)(ζ) characterize thecomplex surface structure, but are independent of surface height at thelocation corresponding to the signal, which is set to zero. In preferredembodiments, the functions m_(temp) ^(j)(ζ) and φ_(temp) ^(j)(ζ) alsoaccount for systematic contributions from the interferometer. We thenuse a complex representation for the template pattern:Ī _(temp) ^(j)(ζ)=m _(temp) ^(j)(ζ)exp[i(K ₀ζ+φ_(temp) ^(j)(ζ))]  (109).We further use a window function to select a particular portion of thecomplex template function:

$\begin{matrix}{{w(\zeta)} = \left\{ \begin{matrix}1 & {{{for}\mspace{14mu}\zeta_{start}} \leq \zeta \leq \zeta_{stop}} \\0 & {otherwise}\end{matrix} \right.} & (110)\end{matrix}$Ĩ _(pat) ^(k)(ζ)=w(ζ)Ī _(temp) ^(k)(ζ)  (111)

For example, an appropriate window might be

$\begin{matrix}{{\zeta_{start} = {- \frac{\Delta\zeta}{2}}}{\zeta_{stop} = {+ \frac{\Delta\;\zeta}{2}}}} & (112)\end{matrix}$where the window width Δζ could be set by hand.

Now that we have an interference pattern template Ĩ_(pat) ^(j) we areready to use it for comparison to an actual data set. In preparation forthis, it will be handy to generate a complex signal Ĩ_(ex) starting froma real experimental data setI _(ex)(ζ,x)=DC _(ex)(x)+ . . . AC _(ex)(x)m _(ex) [ζ−h_(ex)(x)]cos{−[ζ−h _(ex)(x)]K ₀+φ_(ex) [ζ−h _(ex)(x)]}.  (113)The Fourier transform of this signal isq _(ex)(K,x)=FT{I _(ex)(ζ,x)}  (114)q _(ex)(K,x)=δ(K)DC _(ex)(x))+½AC _(ex)(x)[G* _(ex)(−K−K ₀ ,x)+G_(ex)(K−K ₀ ,x)]  (115)whereG _(ex)(K)=FT{m _(ex)(ζ)exp[iφ _(ex)(ζ)]}exp[iKh _(ex)(x)].  (116)We then construct a partial spectrum from the positive-frequency portionof the spectrum:{tilde over (q)} _(ex)(K)=AC _(ex)(x)G _(ex)(K−K ₀ ,x).  (117)The inverse transform is thenĨ _(ex)(ζ)=FT ⁻¹ {{tilde over (q)} _(ex)(K)}  (118)Ĩ _(ex)(ζ,x)=AC _(ex)(x)m _(ex) [ζ−h _(ex)(x)]exp{−i[ζ−h _(ex)(x)]K ₀+iφ _(ex) [ζ−h _(ex)(x)]}  (119)Here, the real part of this complex function Ĩ_(ex) is the originalexperimental data I_(ex). Further, the phase and envelope are separableby simple operations, e.g. we can access the product of the signalstrength AC_(ex)(x) and envelope m_(ex) using the magnitude of thecomplex function Ĩ_(ex):AC _(ex)(x)m _(ex) [ζ−h _(ex)(x)]=|Ĩ _(ex)(ζ,x)|.  (120)According to the underlying theory of the technique, we expect at leasta meaningful portion of m_(ex) to have the same general shape asm_(temp) ^(j) for the correct model, the only difference being thelinear offset h_(ex) and the scaling factor AC_(ex)(x). We also expectthe difference between the experimental and interference patterntemplate phase offsets φ_(ex),φ_(pat) ^(j), respectively, to be linearlyproportional to the height h_(ex), for the correct model.

The task at hand is to locate a specific signal pattern represented bythe interference pattern template Ĩ_(pat) ^(j), within an experimentaldata set Ĩ_(ex), and determine how well of a match there is for each ofthe different models j. In what follows, we shall drop the index j, andnote the matching analysis proceeds for each of the models.

The first step is to find the scan position ζ_(best) for which theshapes of the envelopes m_(ex), m_(pat) and φ_(ex), φ_(pat) are bestmatched. A viable approach is a merit function based on the normalizedcorrelation of the interference pattern template with the signal withina segment of the scan defined by the window w:

$\begin{matrix}{{{\Pi\left( {\zeta,x} \right)} = \frac{{{\overset{\sim}{I}\left( {\zeta,x} \right)}}^{2}}{\left\langle m_{pat}^{2} \right\rangle\left\langle {{{\overset{\sim}{I}}_{ex}\left( {\zeta,x} \right)}}^{2} \right\rangle}}{where}} & (121) \\{{\overset{\sim}{I}\left( {\zeta,x} \right)} = {\frac{1}{N}{\int_{- \infty}^{\infty}{{{\overset{\sim}{I}}_{pat}^{*}\left( \hat{\zeta} \right)}{{\overset{\sim}{I}}_{ex}\left( {{\zeta + \hat{\zeta}},x} \right)}{{\mathbb{d}\hat{\zeta}}.}}}}} & (122)\end{matrix}$is the complex correlation function and

$\begin{matrix}{\left\langle m_{pat}^{2} \right\rangle = {\frac{1}{N}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{I}}_{pat}\left( \hat{\zeta} \right)}}^{2}{\mathbb{d}\hat{\zeta}}}}}} & (123) \\{\left\langle {{{\overset{\sim}{I}}_{ex}\left( {\zeta,x} \right)}}^{2} \right\rangle = {\frac{1}{N}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{I}}_{ex}\left( {{\zeta + \hat{\zeta}},x} \right)}}^{2}{w\left( \hat{\zeta} \right)}{\mathbb{d}\hat{\zeta}}}}}} & (124)\end{matrix}$are normalizations that make the merit function Π independent ofsignal-strength. Use of the complex conjugate Ĩ*_(pat) of the templatecancels the synchronous linear phase term K₀ζ and maximizes Π for thecase of a match of φ_(ex), φ_(pat). The absolute value ∥ of thecorrelation removes any residual complex phase.

To prevent Π(ζ) from generating false high values or encountering asingularity at low signal levels, it is prudent to add a minimum valueto the denominator, such as<|Ĩ _(ex)(ζ)|² >←<|Ĩ _(ex)(ζ)|²>+MinDenom·max(<|Ĩ _(ex)|²>)  (125)where the max ( ) function returns the maximum value of the signalstrength |Ĩ_(ex)| over the full scan length ζ, and MinDenom is theminimum relative signal strength that we consider valid in the meritfunction search. The value of MinDenom can be hard coded at 5% or someother small value, or left as an adjustable parameter.

The correlation integral Ĩ can also be performed in the frequency domainusing the correlation theorem:Ĩ(ζ)=FT ⁻¹ {{tilde over (q)}* _(pat)(K){tilde over (q)} _(ex)(K)}  (126)where we have made use ofFT{Ĩ* _(pat)(ζ,x)}={tilde over (q)}* _(pat)(−K,x)  (127)where{tilde over (q)} _(pat)(K,x)=FT{Ĩ _(pat)(ζ,x)}.  (128)

A search through Π to find a peak value yields the best match positionζ_(best) and the value of Π is a measure of the quality of the match,ranging from zero to one, with one corresponding to a perfect match. Thepeak value of the merit function is calculated for each of the differentmodels to determine which model is the best match, and then the bestmatch position ζ_(best) for that model gives the surface height.

FIGS. 20-24 illustrate an example of the technique. FIG. 20 shows anactual scanning interferometry signal of a base Si substrate without athin film. FIGS. 21 and 22 show interference template patterns for abare Si substrate and a thin film structure with 1 micron of SiO2 on Si,respectively. FIGS. 23 and 24 show the merit function as a function ofscan positions for template functions in FIGS. 21 and 22, respectively.The merit functions show that the interference template pattern for thebare substrate is a much better match (peak value 0.92) than that forthe thin film template pattern (peak value 0.76) and therefore indicatethat the test sample is a bare substrate. Moreover, the position of thepeak in the merit function for the correct template pattern gives therelative surface height position for the test sample.

We now discussed certain examples of the technique that relatespecifically to under-resolved surface features, such as patternedstructures in front-end semiconductor manufacturing processes.

Although patterned by optical lithography, the front end ortransistor-level semiconductor manufacturing process involves featuresthat are far below the resolution limit of visible-wavelengthmicroscopy. The smallest features, such as the transistor gates, are onthe order of 45 nm wide, whereas the instrument transfer function of atypical scanning interferometer falls to zero for a 400-nm periodicstructures. For example, FIG. 28 shows theoretical and experimentalinstrument transfer function for a white-light interference microscopeusing a 100×, 0.8 NA Mirau objective and incoherent illumination. Theseparation of the gates, shallow trench isolation (STI), wires and viasis often comparable to this lower limit; thus we see some of the surfacestructure, but not all. These under-resolved features cannot, therefore,be measured directly as height objects in the usual way with thewhite-light interference microscope. However, parameter monitoring (e.g.depth and width) of these features is often still possible if weunderstand how height variations below optical resolution affect thegeneration of the scanning interferometry data.

FIG. 29 a illustrates a simple cross-sectional model of a symmetricgrating with unpatterned areas on either side. The y coordinate isparallel to the lines and into the figure, while the x coordinate isfrom left to right. The z=0 vertical position corresponds to the tops ofthe lines. A simple-minded scalar diffraction model and the Abbéprinciple confirms that when the grating linewidth L and spacing W fallbelow the resolution limits implied by FIG. 28, the grating lines blurtogether and the NewView cannot directly measure the lateral dimensionsL and W nor the height H. However, this same scalar analysis shows thatthe apparent height of the etched area would indeed depend on the lineheight, width and spacing, even though the gate lines themselves wouldbe indistinguishable. The exact dependence may be predicted to somedegree by modeling, and refined by empirical evidence.

As noted previously, certain embodiments disclosed herein exploit theobservation that the apparent height of the patterned area is related tothe feature parameters of interest. We generate apparent surface heightprofiles using conventional processing interferometric techniques suchas FDA. For the case of a pitch less than one-half wavelength, theseprofiles do not show the grating lines at all, or at most some echo ofthe lines, with an overall “averaged” height H′ as shown in FIG. 29 b,which shows simulated measured surface profiles superimposed on thegrating structure. Consequently, we measure the height H′ between thearea over the lines and the area over the unpatterned bare substrate.This result may be referenced to the zero-etch height by subtracting theetch depth E to get a number E′=H′−E. Alternatively, if a zero-etchmeasured profile is available as in FIG. 29 c (which likewise shows asimulated measured surface profile superimposed on the gratingstructure), the measured etch depth E′ may be referenced directly to thezero-etch height (this is the more usual measurement scenario). Assumingthat we know ahead of time the lateral dimensions L and W, the correctline height H follows from modeling the sensitivity of the entiremeasurement procedure to the parameter of interest. Alternatively, if weknow H and L, we can infer W or some other parameter such as line shape.

Using more rigorous modeling to consider polarization effects shows thatthe simulated profiles shown in FIGS. 29 b and 29 c representqualitatively the behavior when the polarization is orthogonal to thegate lines (defined as the x direction). The parallel or y polarization,in contrast, is much more strongly influenced by the tops of the lines,in some cases appearing as if there were no spaces at all between thelines. This makes physical sense, in that polarization aligned with thegate lines generates currents that increase the influence of the tops ofthe lines at the expense of the areas between the lines. The sensitivityof the measurement to various structure parameters varies withpolarizations; therefore, it may be possible to isolate specificparameters, such as etch depth, while minimizing sensitivity to others,such as linewidth.

Accordingly, embodiments of this measurement technique include: 1) useof a standard interference microscope with circular polarization andcomparison of the step height to a neighboring, unpatterned area in thefield of view (FOV) of known height relative to the pre-etched silicon;2) as above, but use of x (orthogonal to lines) polarization to improvesensitivity to deep (>20 nm) trenches; 3) use of a linear polarizationand comparison of heights of neighboring areas of orthogonally arrangedgrating lines; 4) comparison of the measured heights for both the x andy polarization states in the same field of view with respect to a commonreference that is not polarization dependent, e.g., a smooth, flat area;and 5) comparison of the measured heights for both the x and ypolarization states in the same field of view with respect to eachother, for example by simultaneous capture of data for the twopolarizations, or by interfering the polarization states directly. Thisapproach may obviate any need for a separate reference on the objectsurface.

It should also be noted that the basic measurement principle is notlimited to gratings, but may be extended to other structures.

Although a scalar or Abbé model provides some important insight into thebasic measurement problem, quantitative analysis for a complex(multi-material), under-resolved surface structure can benefit from morerigorous modeling of the interferometry signals by solving Maxwell'sEquations over a range of bandwidths and angles of incidence, such asthe 2D approach based on the Rayleigh Hypothesis or the more rigorousRCWA approach, both of which were described above.

For example, the 2D Rayleigh approach explains the inversion resultshown in FIG. 27 for circularly-polarized light incident on samplewafers on patterned structures. Specifically, in unpolarized orcircularly polarized light, there is a competition between the x and ypolarization states that leads to an inverse correlation. The inversionis attributable to the high sensitivity to etch depth of the xpolarization reflected intensity, causing a modulation in thecontribution of the x polarization compared to the y polarization.Because these two polarizations report different depths, the balancebetween their reflected strengths can lead to this nonlinear correlationbetween the apparent grating modulation depth and the actual gratingmodulation depth. The 2D Rayleigh calculation showed that such aninversion could take place with under-resolved grating lines.

In another example, a rigorous RCWA approach was used to model a puresilicon 5-line grating without top film layers, with lines having awidth W=120 nm and pitch L=320 nm. A schematic of the grating is shownin FIG. 30 a. The output of the simulation, shown in FIG. 30 b, is asimulated SWLI signal (z-direction) for each of a line of pixelsα-direction). FIG. 30 b shows the results only for y polarization,parallel to the lines, which is more sensitive to the tops of the linesthan the areas or trenches between the lines. If we look at the outputfor a single pixel as in FIGS. 31 a and 31 b for the x and ypolarizations, respectively, we recognize the familiar white-lightinterference pattern, approximated by a carrier fringe pattern modulatedby an envelope or fringe contrast function. Noteworthy is the somewhatdistorted x-polarization signal (FIG. 31 a), which reveals a competitionbetween the unresolved tops of the grating lines and the trenches inbetween, leading to a weakened signal with envelope structure and ageneral shift towards the left, which corresponds to lower heights.

An FDA analysis of these signals generates the apparent surface profilesshows in FIGS. 32 a and 32 b for the x-(orthogonal to grating lines) andy-polarizations (parallel to grating lines), respectively. To relatethis to a step height measurement, we compare the center pixel to anedge pixel to determine H′, the edge pixel presumably marking the trueetch depth E of the substrate. The measured substrate etch depth is thenE′=H′−E. This is the value that we would measure when referencing to anunetched substrate area. We rely on modeling or empirical data totranslate the apparent etch depth to the actual depth.

FIGS. 33 a and 33 b show the results of the RCWA analysis for theapparent etch depth E′=H′−E as extracted from simulated scanninginterferometry data as a function of different actual etch depth E forthe y-polarization (parallel to grating lines) and x-(orthogonal tograting lines), respectively. The E′=E line is for reference. FIGS. 34 aand 34 b show the corresponding signal strength for the different actualetch depths E for the y-polarization (parallel to grating lines) andx-(orthogonal to grating lines), respectively. The results show theexpected behavior for the two polarization states. Clearly thepreferable configuration for etch depth sensitivity is the xpolarization, assuming that we can stay clear of the reflected intensityminimum at 145 nm etch depth shown in FIG. 34 b.

The difference between the measured height E′ and etch depth E is insome sense the bias or offset resulting from the unresolved gratinglines, including the top film layers. In the ideal case, the instrumentwould simply disregard the presence of the lines, assuming that thesubstrate etch depth were the only parameter of interest. FIG. 35 is analternative graphing of the same data as in FIG. 33 b, showing that forthe x polarization, the bias is modest and does not change rapidly withetch depth. Specifically, it shows the measurement bias or offset E−E′as a function of the measured step height −E′ for the silicon grating.

FIG. 36 shows the expected RCWA results for circularly polarized lightfor a pure Si grating. Specifically, it shows the RCWA predictions forthe measured etch depth E′=H′−E as a function of the actual etch depth Efor the silicon grating, with circularly polarized light, in comparisonwith a one-to-one correspondence line. Of note is the generally poorresponse of the system, and in particular, the inverse correlationbetween measured and actual etch depth between 60 and 170 nm. This isthe same phenomenon predicted from the Rayleigh hypothesis and shown inFIG. 27 and described above. Above 100 nm etch, the measured depth isnegative, meaning that the etched silicon appears to rise above theunetched silicon. The area looks like a protrusion rather than a recess.

FIGS. 37 a and 37 b show the RCWA predictions for the measured(apparent) etch depth E′=H′−E as a function of the feature width W foran etch depth E=100 nm and pitch L=320 nm, for the five-line silicongrating for y- and x-polarization, respectively. The solid line marksthe etch depth in both graphs, for reference. The graphs shows someinteresting behavior, including relative insensitivity to linewidth overbroad ranges. These results imply that for linewidths between 100 and180 nm, for example, the x-polarization measurement is far moresensitive to etch depth than to linewidth, at least for this combinationof parameters (compare with FIG. 33). The y polarization, however, showsa greater sensitivity to linewidth than to etch depth, although thesensitivity is weak in either case.

Qualitatively the reflectivity minimum in FIG. 33 b and the inversecorrelation in FIG. 36 can be understood as resulting from the situationwhere the depth of the grating corresponds causes a phase flip relatedto the quarter-wave anti-reflection condition. Thus, if it is desired tomove to a more linear portion of the correlation curve, the wavelengthof the light used in the interferometer can be adjusted relative to thequarter-wavelength condition, where the quarter-wavelength condition ismet when the product of the grating depth and the index of the materialbetween the lines (e.g., for the present structure, n=1 for air) equalsthe sum of a quarter of the light wavelength and any integer multiple(including zero) of half the light wavelength. On the other hand, thepresence of a reflectivity minimum (or more generally, reflectivityinformation derived from the interferometry signal strength) can be usedalone, or in conjunction with the apparent surface profile, as theinformation that is compared to expected information for differentmodels of the sample to determine information about the under-resolvedsurface features.

While the above example pertained specifically to a patterned structurein one-dimension, the general principles of the technique can beextended to other types of under-resolved surface patterns, such asdiscrete step heights and two-dimensional patterned structures. Theunderlying principle is that even though such surface features may beunder-resolved they do contribute to both the interferometry signal forindividual pixels and collective information extracted frominterferometry signals from multiple pixels. Thus, experimental derivedinformation can be compared to models that are parameterized bydifferent values for under-resolved features of interest, with the bestcomparison yielding the corresponding value of the feature of interest.

Furthermore, in other examples, the different portions of the gratingmay have different modulations depths or may be formed over differentmaterial compositions. For example, a grating structure can be formed ona silicon substrate that includes regions of silicon dioxide.Furthermore, the regions of the grating between the silicon dioxide canbe etched to different depths. In such embodiments, the information thatis compared to the models can be, for example, differences in apparentsurface height between different resolved regions of the grating. Inother words, for example, although the individual grating lines may notbe resolved, the portion of the grating formed over the silicon dioxidemay be distinguishable from other portions of the grating.

FIG. 38 shows a schematic diagram of how various components ininterferometry system 900 used to generate the interferometry signalscan be automated under the control of electronic processor 970, which,in the presently described embodiment, can include an analyticalprocessor 972 for carrying out mathematical analyses (e.g., comparisonto the model libraries), device controllers 974 for controlling variouscomponents in the interferometry system, a user interface 976 (e.g., akeyboard and display), and a storage medium 978 for storing information(e.g., library models and calibration information), data files, and/orautomated protocals. Interferometry system 900 generally directs testlight 922 to test object 926, which is secured to mount 940 on a stage950.

First, the system can include a motorized turret 910 supporting multipleobjectives 912 and configured to introduce a selected objective into thepath of the input light. One or more of the objectives can beinterference objectives, with the different interference objectivesproviding different magnifications. Furthermore, in certain embodiments,the interference objectives can include one (or more) polarizationelements (e.g., a linear polarizer) attached to it (for example, toilluminate a grating pattern with light polarized orthogonal to thegrating lines). In such cases, it the orientation of the polarizationelement can be also be under automated control so as to, for example,align the polarization of the illumination light with respect thegrating lines of a patterned structure. Moreover, one or more of theobjectives can be a non-interferometric objective (i.e., one without areference leg), each with a different magnification, so that system 900can also operate in a conventional microscope mode for collectingoptical images of the test surface. Turret 910 is under the control ofelectronic processor 970, which selects the desired objective accordingto user input or some automated protocol.

Next, the system includes a motorized stage 920 (e.g., a tube lensholder) for supporting relay lenses 936 and 938. Such a stage can beused for selecting between a profiling mode, such as that generallycontemplated in the present application, or an ellipsometry orreflectometry mode in which the pupil plane is imaged to the detector,as described in U.S. Patent Publication No. US-2006-0158659-A1 entitled“INTERFEROMETER FOR DETERMINING CHARACTERISTICS OF AN OBJECT SURFACE” byColonna de Lega et al., the contents of which are incorporated byreference. Motorized stage 920 is under the control of electronicprocessor 970, which selects the desired relay lens according to userinput or some automated protocol. In other embodiments, in which atranslation stage is moved to adjust the position of the detector toswitch between the first and second modes, the translation is undercontrol of electronic processor. Furthermore, in those embodiments withmultiple detection channels, each detector is coupled to the electronicprocessor 970 for analysis.

Furthermore, the system can include motorized apertures 930 and 932under control of electronic processor 970 to control the dimensions ofthe field stop and aperture stop, respectively. Again the motorizedapertures are under the control of electronic processor 970, whichselects the desired settings according to user input or some automatedprotocol.

Also, translation stage 980, which is used to vary the relative opticalpath length between the test and reference legs of the interferometer,is under the control electronic processor 970. The translation stage canbe coupled to adjust the position of the interference objective relativeto a mount 940 for supporting test object 926. Alternatively, in furtherembodiments, the translation stage can adjust the position of theinterferometry system as a whole relative to the mount, or thetranslation stage can be coupled to the mount, so it is the mount thatmoves to vary the optical path length difference.

Furthermore, a lateral translation stage 950, also under the control ofelectronic processor 970, can be coupled to the mount 940 supporting thetest object to translate laterally the region of the test surface underoptical inspection. In certain embodiments, translation stage 950 canalso orient mount 940 (e.g. provide tin and tilt) so as to align thetest surface normal to the optical axis of the interference objective.

Finally, an object handling station 960, also under control ofelectronic processor 970, can be coupled to mount 940 to provideautomated introduction and removal of test samples into system 900 formeasurement. For example, automated wafer handling systems known in theart can be used for this purpose. Furthermore, if necessary, system 900and object handling system can be housed under vacuum or clean roomconditions to minimize contamination of the test objects.

The resulting system provides great flexibility for providing variousmeasurement modalities and procedures. For example, the system can firstbe configured in the microscope mode with one or more selectedmagnifications to obtain optical images of the test object for variouslateral positions of the object. Such images can be analyzed by a useror by electronic processor 970 (using machine vision techniques) toidentify certain regions (e.g., specific structures or features,landmarks, fiducial markers, defects, etc.) in the object. Based on suchidentification, selected regions of the sample can then be studied inthe ellipsometry mode to determine sample properties (e.g., refractiveindex, underlying film thickness(es), material identification, etc.).

When used in conjunction with automated object handling system 960, themeasurement procedure can be repeated automatically for a series ofsamples. This could be useful for various process control schemes, suchas for monitoring, testing, and/or optimizing one or more semiconductorprocessing steps.

For example, the system can be used in a semiconductor process for toolspecific monitoring or for controlling the process flow itself. In theprocess monitoring application, single/multi-layer films are grown,deposited, polished, or etched away on unpatterned Si wafers (monitorwafers) by the corresponding process tool and subsequently the thicknessand/or optical properties are measured using the interferometry systemdisclosed herein. The average, as well as within wafer uniformity, ofthickness (and/or optical properties) of these monitor wafers are usedto determine whether the associated process tool is operating withtargeted specification or should be retargeted, adjusted, or taken outof production use.

In the process control application, latter single/multi-layer films aregrown, deposited, polished, or etched away on patterned Si, productionwafers by the corresponding process tool and subsequently the thicknessand/or optical properties are measured with the interferometry systemdisclosed herein (for example, by using the ellipsometry mode, theprofiling mode, or both). Production measurements used for processcontrol typical include a small measurement site and the ability toalign the measurement tool to the sample region of interest. This sitemay consists of multi-layer film stack (that may itself be patterned)and thus requires complex mathematical modeling in order to extract therelevant physical parameters. Process control measurements determine thestability of the integrated process flow and determine whether theintegrated processing should continue, be retargeted, redirected toother equipment, or shut down entirely.

Specifically, for example, the interferometry system disclosed hereincan be used to monitor the following equipment: diffusion, rapid thermalanneal, chemical vapor deposition tools (both low pressure and highpressure), dielectric etch, chemical mechanical polishers, plasmadeposition, plasma etch, lithography track, and lithography exposuretools. Additionally, the interferometry system disclosed herein can beused to control the following processes: trench and isolation,transistor formation, as well as interlayer dielectric formation (suchas dual damascene). The methods and systems described above can beparticularly useful in semiconductor applications. Additionalembodiments of the invention include applying any of the measurementtechniques described above to address any of the semiconductorapplications described below, and systems for carrying out both themeasurement techniques and the semiconductor applications.

It is presently of considerable interest in the semiconductor industryto make quantitative measurements of surface topography. Due to thesmall size of typical chip features, the instruments used to make thesemeasurements typically must have high spatial resolution both paralleland perpendicular to the chip surface. Engineers and scientists usesurface topography measuring systems for process control and to detectdefects that occur in the course of manufacturing, especially as aresult of processes such as etching, polishing, cleaning and patterning.

For process control and defect detection to be particularly useful, asurface topography measuring system should have lateral resolutioncomparable to the lateral size of typical surface features, and verticalresolution comparable to the minimum allowed surface step height.Typically, this requires a lateral resolution of less than a micron, anda vertical resolution of less than 1 nanometer. It is also preferablefor such a system to make its measurements without contacting thesurface of the chip, or otherwise exerting a potentially damaging forceupon it, so as to avoid modifying the surface or introducing defects.Further, as it is well-known that the effects of many processes used inchip making depend strongly on local factors such as pattern density andedge proximity, it is also important for a surface topography measuringsystem to have high measuring throughput, and the ability to sampledensely over large areas in regions which may contain one or manysurface features of interest.

It is becoming common among chip makers to use the so-called ‘dualdamascene copper’ process to fabricate electrical interconnects betweendifferent parts of a chip. This is an example of a process which may beeffectively characterized using a suitable surface topography system.The dual damascene process may be considered to have five parts: (1) aninterlayer dielectric (ILD) deposition, in which a layer of dielectricmaterial (such as a polymer, or glass) is deposited onto the surface ofa wafer (containing a plurality of individual chips); (2) chemicalmechanical polishing (CMP), in which the dielectric layer is polished soas to create a smooth surface, suitable for precision opticallithography, (3) a combination of lithographic patterning and reactiveion etching steps, in which a complex network is created comprisingnarrow trenches running parallel to the wafer surface and small viasrunning from the bottom of the trenches to a lower (previously defined)electrically conducting layer, (4) a combination of metal depositionsteps which result in the trenches and vias being over-filled withcopper, and (5) a final chemical mechanical polishing (CMP) step inwhich the excess copper is removed, leaving a network of copper filledtrenches (and possibly vias) surrounded by dielectric material.

Typically the thickness of the copper in the trench areas (i.e., thetrench depth), and the thickness of the surrounding dielectric lie in arange of 0.2 to 0.5 microns. The width of the resulting trenches may bein a range of from 100 to 100,000 nanometers, and the copper regionswithin each chip may in some regions form regular patterns such asarrays of parallel lines, and in others they may have no apparentpattern. Likewise, within some regions the surface may be denselycovered with copper regions, and in other regions, the copper regionsmay be sparse. It is important to appreciate that the polishing rate,and therefore the remaining copper (and dielectric) thickness afterpolishing, depends strongly and in a complex manner on the polishingconditions (such as the pad pressure and polishing slurry composition),as well as on the local detailed arrangement (i.e., orientation,proximity and shape) of copper and surrounding dielectric regions.

This ‘position dependent polishing rate’ is known to give rise tovariable surface topography on many lateral length scales. For example,it may mean that chips located closer to the edge of a wafer onaggregate are polished more rapidly than those located close to thecenter, creating copper regions which are thinner than desired near theedges, and thicker than desired at the center. This is an example of a‘wafer scale’ process nonuniformity—i.e., one occurring on length scalecomparable to the wafer diameter. It is also known that regions whichhave a high density of copper trenches polish at a higher rate thannearby regions with low copper line densities. This leads to aphenomenon known as ‘CMP induced erosion’ in the high copper densityregions. This is an example of a ‘chip scale’ processnon-uniformity—i.e., one occurring on a length scale comparable to (andsometimes much less than) the linear dimensions of a single chip.Another type of chip scale nonuniformity, known as ‘dishing’, occurswithin single copper filled trench regions (which tend to polish at ahigher rate than the surrounding dielectric material). For trenchesgreater than a few microns in width dishing may become severe with theresult that affected lines later exhibit excessive electricalresistance, leading to a chip failure.

CMP induced wafer and chip scale process nonuniformities are inherentlydifficult to predict, and they are subject to change over time asconditions within the CMP processing system evolve. To effectivelymonitor, and suitably adjust the process conditions for the purpose ofensuring that any nonuniformities remain within acceptable limits, it isimportant for process engineers to make frequent non-contact surfacetopography measurements on chips at a large number and wide variety oflocations. This is possible using embodiments of the interferometrytechniques described above.

The interferometer embodiments described above include an interferenceobjective of the Mirau-type and of the Linnik-type. In the Mirau, thebeam splitter in the interference objective directs the reference lightback along the optical axis for the test light. In the Linnik-type, thebeam splitter is positioned prior to the objective lens for the testsurface (with respect to the input light) and directs the test andreference light along different paths. A separate objective lens is usedto focus the reference light to the reference lens. In other words, thebeam splitter separates the input light into the test and referencelight, and separate objective lenses then focus the test and referencelight to respective test and reference surfaces. Ideally the twoobjective lenses are matched to one another so that the test andreference light have similar aberrations and optical paths.

In other embodiments, the interferometry system can instead use adifferent type of interference objective, such as a Michelson objective,in which the beam splitter directs the reference light away from theoptical axis of the test light (e.g., the beam splitter can be orientedat 45 degrees to the input light so the test light and reference travelat right angles to one another). In such cases, the reference surfacecan be positioned outside of the path of the test light.

Additional interferometer configurations are also possible. For example,the system can be configured to collect test light that is transmittedthrough the test sample and then subsequently combined with referencelight. For such embodiments, for example, the system can implement aMach-Zehnder interferometer with dual microscope objectives on each leg.

The light source in the interferometer may be any of: an incandescentsource, such as a halogen bulb or metal halide lamp, with or withoutspectral bandpass filters; a broadband laser diode; a light-emittingdiode; a combination of several light sources of the same or differenttypes; an arc lamp; any source in the visible spectral region; anysource in the IR spectral region, particularly for viewing roughsurfaces & applying phase profiling; and any source in the UV spectralregion, particularly for enhanced lateral resolution. For broadbandapplications, the source preferably has a net spectral bandwidth broaderthan 5% of the mean wavelength, or more preferably greater than 10%,20%, 30%, or even 50% of the mean wavelength. For tunable, narrow-bandapplications, the tuning range is preferably broad (e.g., greater than50 nm, greater than 100 nm, or greater than even 200 nm, for visiblelight) to provide information over a wide range of wavelengths, whereasthe spectral width at any particular setting is preferable narrow, tooptimize resolution, for example, as small as 10 nm, 2 nm, or 1 nm. Thesource may also include one or more diffuser elements to increase thespatial extent of the input light being emitted from the source.

Furthermore, the various translations stages in the system, such astranslation stage 150, may be: driver by any of a piezo-electric device,a stepper motor, and a voice coil; implemented opto-mechanically oropto-electronically rather than by pure translation (e.g., by using anyof liquid crystals, electro-optic effects, strained fibers, and rotatingwaveplates) to introduce an optical path length variation; any of adriver with a flexure mount and any driver with a mechanical stage, e.g.roller bearings or air bearings. As noted above, while thephase-shifting for the scanning interferometry signal is often done byusing a mechanical translation stage, it is also possible to vary thephase between the test and reference legs of the interferometer byvarying the wavelength of the source when there is a non-zero opticalpath length difference between the test and reference legs.

The electronic detector can be any type of detector for measuring anoptical interference pattern with spatial resolution, such as amulti-element CCD or CMOS detector.

Any of the computer analysis methods described above can be implementedin hardware or software, or a combination of both. The methods can beimplemented in computer programs using standard programming techniquesfollowing the method and figures described herein. Program code isapplied to input data to perform the functions described herein andgenerate output information. The output information is applied to one ormore output devices such as a display monitor. Each program may beimplemented in a high level procedural or object oriented programminglanguage to communicate with a computer system. However, the programscan be implemented in assembly or machine language, if desired. In anycase, the language can be a compiled or interpreted language. Moreover,the program can run on dedicated integrated circuits preprogrammed forthat purpose.

Each such computer program is preferably stored on a storage medium ordevice (e.g., ROM or magnetic diskette) readable by a general or specialpurpose programmable computer, for configuring and operating thecomputer when the storage media or device is read by the computer toperform the procedures described herein. The computer program can alsoreside in cache or main memory during program execution. The analysismethod can also be implemented as a computer-readable storage medium,configured with a computer program, where the storage medium soconfigured causes a computer to operate in a specific and predefinedmanner to perform the functions described herein. While the

1. A method comprising: comparing information derivable from multipleinterferometry signals corresponding to different surface locations of atest object to information corresponding to multiple models of the testobject, wherein the multiple models are parameterized by a series ofcharacteristics that relate to one or more under-resolved lateralfeatures of the test object; and outputting information about theunder-resolved surface feature based on the comparison.
 2. The method ofclaim 1, wherein the one or more under-resolved lateral features of thetest object correspond to one or more of a pitch, a modulation depth,and an element width for an under-resolved patterned lateral structureon the test object.
 3. The method of claim 1, wherein the one or moreunder-resolved lateral features of the test object correspond to atleast a modulation depth for an under-resolved patterned lateralstructure on the test object.
 4. The method of claim 3, wherein theseries of characteristics are different values for the modulation depth.5. The method of claim 4, wherein the multiple models are represented bya correlation that maps possible outcomes for the information derivablefrom the multiple interferometry signals to corresponding ones of thedifferent values for the modulation depth, and the comparing comprisesdetermining which of the different values for the modulation depth bestcorresponds to the information derivable from the multipleinterferometry signals.
 6. The method of claim 3, wherein the modulationdepth can be expressed relative to bias offset value.
 7. The method ofclaim 2, wherein at least some of the interferometry signals are derivedfrom an illumination of the test object with the illumination having apolarization that is oriented with respect to the elements of thepatterned lateral structure.
 8. The method of claim 7, wherein thepolarization is a linear polarization aligned orthogonal to the lengthof the individual elements that define the patterned lateral structure.9. The method of claim 1, wherein the one or more under-resolved lateralfeatures of the test object correspond to one or more of a height and aposition of a step on the test object.
 10. The method of claim 9,wherein the series of characteristics comprise different values for theheight or position of the step height.
 11. The method of claim 1,wherein the information derivable from the multiple interferometrysignals comprises one or more values extracted from a height profile forthe test object derived from the multiple interferometry signals,wherein the under-resolved surface feature is obscured or does notappear in the extracted height profile.
 12. The method of claim 11,wherein the test object comprises a patterned lateral structure whoseindividual element are obscured or do not appear in the extractedsurface height profile.
 13. The method of claim 12, wherein theinformation derivable from the multiple interferometry signals is avalue for a height for a collection of under-resolved elements in thepatterned lateral structure extracted from the height profile.
 14. Themethod of claim 13, wherein the information about the under-resolvedsurface feature corresponds to one or more of a modulation depth and, anelement width for the patterned lateral structure.
 15. The method ofclaim 13, wherein the different surface locations for the interferometrysignals include a reference portion of the test object providing areference height value for the extracted height profile.
 16. The methodof claim 15, wherein the test object is etched to produce the patternedstructure, and wherein the reference portion of the test object is aportion of the test object known to not be etched.
 17. The method ofclaim 12, wherein at least some of the interferometry signals from whichthe height profile is determined are derived from an illumination of thetest object with the illumination having a polarization that is orientedwith respect to the elements of the patterned lateral structure.
 18. Themethod of claim 17, wherein the polarization is a linear polarizationaligned orthogonal to the length of the individual elements that definethe patterned lateral structure.
 19. The method of claim 11, wherein theheight profile is obtained from a frequency domain analysis of theinterferometry signals.
 20. The method of claim 11, wherein the heightprofile is obtained from a relative position of a coherence peak in eachinterferometry signal.
 21. The method of claim 1, wherein theunder-resolved lateral features of the test object have a feature sizesmaller than 400 nm.
 22. The method of claim 1, wherein theunder-resolved lateral features of the test object have a feature sizesmaller than 200 nm.
 23. The method of claim 1, wherein theunder-resolved lateral features of the test object have a feature sizesmaller than 100 nm.
 24. The method of claim 1, wherein the models aregenerated computationally using rigorous coupled wave analysis (RCWA).25. The method of claim 1, wherein the models are generated empiricallyfrom test objects having known properties.
 26. The method of claim 1,wherein the information about the under-resolved surface feature isoutputted to a user.
 27. The method of claim 1, wherein the informationabout the under-resolved surface feature is outputted to an automatedprocess control system for semiconductor manufacturing.
 28. The methodof claim 1, wherein the interferometry signals are scanninginterferometry signals.
 29. The method of claim 28, wherein the scanninginterferometry signal is produced by imaging test light emerging fromthe test object to interfere with reference light on a detector, andvarying an optical path length difference from a common source to thedetector between interfering portions of the test and reference light,wherein the test and reference light are derived from the common source,and wherein the scanning interferometry signal corresponds to aninterference intensity measured by the detector as the optical pathlength difference is varied.
 30. The method of claim 29, furthercomprising producing the scanning interferometry signal.
 31. The methodof claim 29, wherein the test and reference light have a spectralbandwidth greater than 5% of a central frequency for the test andreference light.
 32. The method of claim 31, wherein the common sourcehas a spectral coherence length, and the optical path length differenceis varied over a range larger than the spectral coherence length toproduce the scanning interferometry signal.
 33. The method of claim 29,wherein optics used to direct test light onto the test object and imageit to the detector define a numerical aperture for the test lightgreater than 0.8.
 34. The method of claim 33, wherein the common, sourceis a spatially extended source.
 35. A method for determining one or morespatial properties of a grating structure on a test object, wherein thegrating structure comprises line elements having widths smaller than 400nm so that the line elements are not fully resolved by an interferencemicroscope, the method comprising: determining an apparent height for acollection of at least some of the grating lines from interferencesignals at different locations of the test object measured by theinterference microscope; providing an expected response for theinterference microscope for different possible values for the propertiesof the grating structure, wherein the expected response includescontributions from under-resolved line elements of the gratingstructure; comparing the apparent height to the expected response forthe different possible values to determine information about the spatialproperties of a grating structure; and outputting the determinedinformation about the spatial properties of the grating structure. 36.The method of claim 35, wherein the apparent height is determined withreference to a reference portion of the test object.
 37. The method ofclaim 35, wherein the interference microscope illuminates the gratingstructure with light polarized orthogonal to length of the individualgrating lines when determining the apparent height.
 38. The method ofclaim 35, wherein the determined information about the spatialproperties of the grating structure corresponds to a modulation depthfor the grating structure.
 39. The method of claim 35, wherein thegrating structure is a series of periodically spaced lines formed atleast in part by etching portions of the test object between the lines.40. The method of claim 35, wherein the interference signals arescanning interference signals produced by imaging test light emergingfrom the test object to interfere with reference light on a detector,and varying an optical path length difference from a common source tothe detector between interfering portions of the test and referencelight, wherein the test and reference light are derived from the commonsource, and wherein each scanning interferometry signal corresponds toan interference intensity measured by the detector as the optical pathlength difference is varied.
 41. The method of claim 40, wherein theoptical path length difference is varied over a range larger than thecoherence length of the interference microscope.
 42. A methodcomprising: determining one or more apparent properties of a testsurface from interferometry signals produced by an interferometrysystem; comparing the apparent properties determined from theinterferometry signals to an expected response for the interferometrysystem for different possible values for one or more under-resolvedfeatures of the test surface; and outputting information about the oneor more under-resolved features of the test surface based on thecomparison.
 43. The method of claim 42, wherein the interferometrysystem is a scanning interferometry system.
 44. The method of claim 42,wherein the apparent properties of the test surface are determining fromthe interferometry signal based on variations in any of interferencephase, interference contrast, and surface reflectivity.
 45. The methodof claim 42, wherein the expected response is calculated for variationsin one or more of surface height and surface composition.
 46. The methodof claim 42, wherein the test surface comprises a patterned structurehaving a modulation depth, a periodicity, and elements having a width,and wherein the expected response is calculated for variations in one ormore of the modulation depth, the periodicity, and element width. 47.The method of claim 46, wherein the expected response is calculated forvariations in the modulation depth.
 48. The method of claim 47, whereinthe correspondence between the actual modulation depth and the apparentmodulation calculated for the expected response includes a positivecorrelation over a first range of actual modulation depths and anegative correlation over a second range of actual modulation depths.49. The method of claim 42, using the information about the one or moreunder-resolved features to monitor one or more semiconductor processingsteps.
 50. The method of claim 49, wherein the semiconductor processingsteps include any of isolation patterning and etch, poly-silicon gateelectrode patterning and etch, source/drain etch and deposition, andmetallization patterning, etch, and polish processing.
 51. Apparatuscomprising: a computer readable medium having a program that causes aprocessor in a computer to compare information derivable from multipleinterferometry signals corresponding to different surface locations of atest object to information corresponding to multiple models of the testobject, wherein the multiple models are parameterized by a series ofcharacteristics that relate to one or more under-resolved lateralfeatures of the test object, and output information about theunder-resolved surface feature based on the comparison.
 52. Apparatuscomprising: an interferometry system configured to produce multipleinterferometry signals corresponding to different surface locations of atest object; and an electronic processor coupled to the interferometrysystem to receive the interferometry signals and programmed to compareinformation derivable from the multiple interferometry signals toinformation corresponding to multiple models of the test object, whereinthe multiple models are parameterized by a series of characteristicsthat relate to one or more under-resolved lateral features of the testobject, and output information about the under-resolved surface featurebased on the comparison.
 53. An apparatus for determining one or morespatial properties of a grating structure on a test object, wherein thegrating structure comprises line elements having widths smaller than 400nm so that the line elements are not hilly resolved by an interferencemicroscope, the apparatus comprising: a computer readable medium havinga program that causes a processor in a computer to: 1) determine anapparent height for a collection of at least some of the grating linesfrom interference signals at different locations of the test objectmeasured by the interference microscope; 2) provide an expected responsefor the interference microscope for different possible values for theproperties of the grating structure, wherein the expected responseincludes contributions from under-resolved line elements of the gratingstructure; 3) compare the apparent height to the expected response forthe different possible values to determine information about the spatialproperties of a grating structure; and 4) output the determinedinformation about the, spatial properties of the grating structure. 54.An apparatus for determining one or more spatial properties of a gratingstructure on a test object, wherein the grating structure comprises lineelements having widths smaller than 400 nm so that the line elements arenot fully resolved by an interference microscope, the apparatus,comprising: the interference microscope; and an electronic processorcoupled to the interference microscope and programmed to: 1) determinean apparent height for a collection of at least some of the gratinglines from interference signals at different locations of the testobject measured by the interference microscope; 2) provide an expectedresponse for the interference microscope for different possible valuesfor the properties of the grating structure, wherein the expectedresponse includes contributions from under-resolved line elements of thegrating structure; 3) compare the apparent height to the expectedresponse for the different possible values to determine informationabout the spatial properties of a grating structure; and 4) output thedetermined information about the spatial properties of the gratingstructure.
 55. Apparatus comprising: a computer readable medium having aprogram that causes a processor in a computer to: 1) determine one ormore apparent properties of a test surface from interferometry signalsproduced by an interferometry system; 2)compare the apparent propertiesdetermined from the interferometry signals to an expected response forthe interferometry system for different possible values of one or moreunder-resolved features of the test surface; and 3) output informationabout the one or more under-resolved features of the test surface basedon the comparison.
 56. Apparatus comprising: an interferometry systemconfigured to produce multiple interferometry signals corresponding todifferent surface locations of a test object; and an electronicprocessor coupled to the interferometry system to receive theinterferometry signals and programmed to: 1) determine one or moreapparent properties of a test surface from the interferometry signals;2) compare the apparent properties determined from the interferometrysignals to an expected response for the interferometry system fordifferent possible values of one or more under-resolved features of thetest surface; and 3) output information about the one or moreunder-resolved features of the test surface based on the comparison.